Things the New Testament church did not manifestly believe or practice, as seen in Scripture

Distinctive Catholic teachings are not manifest in the only wholly inspired substantive authoritative record of what the NT church believed (which is Scripture, in particular Acts through Revelation, which best shows how the NT church understood the gospels).

1. A church that believed that being the instrument and steward of Divine revelation, and inheritor of Divine promises of God's presence and preservation, and having historical decent, makes such the infallible interpreter of Scripture, and without whose sanction one cannot have authority.

They could not believe that, as the church began contrary to the Catholic premise, but in dissent from the very magisterium who sat in the seat of Moses, (Mt. 23:2), Who are Israelites; to whom pertaineth the adoption, and the glory, and the covenants, and the giving of the law, and the service of God, and the promises." (Romans 9:4)  

And they had listened to a holy anointed man in the desert who eat insects, and taught God is able  to raise up children unto Abraham from stones." (Matthew 3:9) And they believed an itinerant Preacher from Galilee who  did not have the sanction of the magisterium, and who rejected Him. (Mk. 11:27-33)  But who established His claims upon Scriptural substantiation, in text and in power, as did the apostles and early church. (Mt. 22:23-45; Lk. 24:27,44; Jn. 5:36,39; Acts 2:14-35; 4:33; 5:12; 15:6-21;17:2,11; 18:28; 28:23; Rm. 15:19; 2Cor. 12:12, etc.)

2. A magisterium whose office was perpetually assured never to err whenever it universally spoke on faith and morals. And which infallibly declared that it was, as per Rome, and led by a supreme pope whose successors as individuals had the same unique charism. 

That an infallible magisterium is necessary to be the steward of Divine revelation and fulfill the promise of preservation is not Scriptural, but is an ecclesiastical extrapolation by Catholics. We know that Acts 15 issued Divine binding judgment because it was solidly based on Scriptural substantiation, and was recorded in Scripture.

3. A pope who was elected by voting, not the OT method of casting lots, and which thus was often the result of included bribery, political intimidation, and centuries of Italian superiority, and immoral men who would be qualified to be or remain a church member, let alone an apostolic successor. "And they prayed, and said, Thou, Lord, which knowest the hearts of all men, shew whether of these two thou hast chosen,” “And they gave forth their lots; and the lot fell upon Matthias; and he was numbered with the eleven apostles. " (Acts 1:24,26)

4. A corporate church that looked to a pope in Rome as its first supreme infallible head to be perpetuated, or any exhortation or command to the churches to submit to Peter as its supreme head, or such a title being give him confirming him as the rock upon which the church was built, as Rome contends (without the “unanimous consent of the fathers”). A most conspicuous absence in the light of the type of cardinal importance Rome shows.

"If one had asked a Christian in the year 100, 200, or even 300 whether the bishop of Rome was the head of all Christians, or whether there was a supreme bishop over all the other bishops and having the last word in questions affecting the whole Church, he or she would certainly have said no." (Jesuit Father theologian, and professor of church history Klaus Schatz, in “Papal Primacy,” p. 3)

Peter was first in rank and in initial use among the 12, and did exercise a general pastoral role, and was first to use the “keys.” Yet Paul could state, "for in nothing am I behind the very chiefest apostles, though I be nothing." (2 Corinthians 12:11) 

The extensive attributes for a “pope Paul” have even been listed in a tongue-in-cheek response to RC attempts to exalt Peter above that which is written. And who listed Peter as second among them who them “which were of reputation,” “who seemed to be pillars.” (Gal. 2:2,9)

As for Peter being rock upon which the church is built, besides that interpretation not having "unanimous consent" of the fathers (which is spun to mean something other than what the term conveys) for what its worth, that the Lord Jesus is the rock of Peter's confession upon which the church is built is one of the most abundantly confirmed teachings in the Bible (petra: Rm. 9:33; 1Cor. 10:4; 1Pet. 2:8; Rev. 4:3) or "stone" (lithos) (Mat. 21:42; Mk.12:10-11; Lk. 20:17-18; Acts. 4:11; Rm. 9:33; Eph. 2:20) as stated by Peter himself. (1Pet. 2:4-8; cf. Dt 32:4, Is. 28:16)

5. The mention of any successors to the original apostles (James: Acts 12:2) besides Judas, he being elected to preserve the foundational twelve apostles, (Acts 1:16-26; cf. Rev. 21:14)

6. A separate sacerdotal class of clergy titled "priests - a word never used distinctly for any pastor, in distinction with the Jews, and the only time it is used for anyone in the NT church it is for the general priesthood of all believers. (1Pt. 2:9) "For this cause left I thee in Crete, that thou shouldest set in order the things that are wanting, and ordain elders in every city, ...For a bishop must be blameless..." (Titus 1:5,7)

7. A clergy that differentiated between bishops and elders, and with grand titles ("Most Reverend Eminence," “Very Reverend,” “Most Illustrious and Most Reverend Lord,” “His Eminence Cardinal,” “The Most Reverend the Archbishop,” etc.) and also made distinct by their ostentatious pompous garb. (Matthew 23:5-7) 

And likewise with a pope enthroned like a Caesar, with his subjects bowing down to him and kissing his feet." "And as Peter was coming in, Cornelius met him, and fell down at his feet, and worshipped him. But Peter took him up, saying, Stand up; I myself also am a man." (Acts 10:25-26)

8. Required (with rare exceptions) clerical celibacy, which presumes all such have that gift. "A bishop then must be blameless, the husband of one wife, ..One that ruleth well his own house, having his children in subjection with all gravity; (For if a man know not how to rule his own house, how shall he take care of the church of God?) " (1 Timothy 3:2,4,5) "For I would that all men were even as I myself. But every man hath his proper gift of God, one after this manner, and another after that. " (1 Corinthians 7:7)

9. A separate class of believers called “saints,” or the mention of the postmortem location of the saints being in purgatory versus with the Lord. (Lk. 24:43; 2Cor. 5:8; Phil. 1:23; 1Thes. 4:17)

10. The practice of praying to the departed, and the hyper exaltation of and devotion to Mary above that which is written; (1Cor. 4:6)

11. The practice of baptizing infants. Infants cannot fulfill the stated requirements for baptism, (Acts 2:38; 8:36,37) and the NT conspicuously provides zero examples of this supposedly critical practice, and where there is any description of household baptisms, what is supported is that the baptized could hear. "But when they believed Philip preaching the things concerning the kingdom of God, and the name of Jesus Christ, they were baptized, both men and women. " (Acts 8:12) "And they spake unto him the word of the Lord, and to all that were in his house. And he took them the same hour of the night, and washed their stripes; and was baptized, he and all his, straightway. " (Acts 16:32-33)

12. A gospel which promoted institutionalized religion, in which one is held to be born again due to the very act of baptism, even though the subject cannot repent and believe, and then is taught they can eventually become good enough to enter Heaven through the power of the church, by taking part in rituals which ex opere operato (by the act itself) convey grace but foster perfunctory professions, and then typically through purgatory, indulgences and offerings to departed souls.

13. A church that conformed to this world in using papal sanctioned physical oppression torture, burning and death to deal with theological dissent. Or who, having lost that power, treats notorious manifestly impenitent public sinners as members in life and in death, in contrast to the NT means of disfellowship and spiritual discipline. “I wrote unto you in an epistle not to company with fornicators ....But them that are without God judgeth. Therefore put away from among yourselves that wicked person. " (1 Corinthians 5:13) "Of whom is Hymenaeus and Alexander; whom I have delivered unto Satan, that they may learn not to blaspheme. " (1 Timothy 1:20)

14. Laws that greatly restricted personal reading of Scripture by laity (contrary to Chrysostom), if able and available, sometimes even outlawing it when it was. "And that from a child thou hast known the holy scriptures, which are able to make thee wise unto salvation through faith which is in Christ Jesus. " (2 Timothy 3:15) "And Paul, as his manner was, went in unto them, and three sabbath days reasoned with them out of the scriptures." (Acts 17:2)

15. A church that sanctioned teaching millions such things as that OT miraculous stories are fables or folktales, and that some historical accounts may not be literally accurate (sermon on the mount, etc.) "For as Jonas was three days and three nights in the whale's belly; so shall the Son of man be three days and three nights in the heart of the earth. " (Matthew 12:40)

16. A church that taught that the deity Muslims worship (not as unknown) is the same as theirs. "But I say, that the things which the Gentiles sacrifice, they sacrifice to devils, and not to God: and I would not that ye should have fellowship with devils. " (1 Corinthians 10:20)

17. The Lord's supper being the means by which souls gain life in them, or not “discerning the body” as referring to the elements of the supper versus members of the church.

Catholics can plead tradition on this, but what so-called "church fathers  believed does not necessarily have any more warrant than some of the unbalanced views on celibacy versus marriage some had, and the wresting of Scripture in trying to support them that even Jerome engaged in. However, one can be a real believer and yet hold to some errors, a belief in the a "real presence" being one of them.  

Finally, there are many more things of contrast that can be added but which are not necessary restricted to Catholicism, and it must be said that the real alternative to Catholicism, that of churches which manifest Scripture (not their church or men) as being supreme and authoritative as the assured word of God, and preaching salvation by grace, also has things that are contrary to the NT church. Primarily these typically are a lack of emphasis on baptism in confessing Christ (not as being necessary to be born again, but which can be the occasion of it), the nature of saving faith as effecting holiness in heart and deed, the laying on of hands in conveyance of the fullness of the Holy Spirit, (Acts 9:17) and Scriptural operation of all gifts, and a central magisterium with apostolic holiness and real spiritual power, and division.

However, overall essential conformity with Scriptural and the working of Holy Spirit is far more manifest in evangelical churches than in Catholicism, which preaches an institutionalized gospel in which souls are imagined to be born again through the act of baptizing, even if they have no repentance or faith, and go on to believe they will finally become good enough to enter Heaven through the power of the church (rituals, purgatory, indulgences, prayers to the departed, etc.). And which can only claim it has apostolic holiness and real spiritual power, and a central magisterium with such, while its foundational premise of perpetual assured infallibility of office is unScriptural. In addition, Catholicism itself bounds with disagreements, and exists as sects and in formal division with more than one central magisterium, each infallibly differing on what Tradition, Scripture and history teach. More here on unity.

The early church did not teach perpetual assured infallibility of office, whereby Roman Catholics find assurance, but both the Lord and His disciples established their claims upon Scriptural substantiation in word and in power. (Mt. 22:23-45; Lk. 24:27,44; Jn. 5:36,39; Acts 2:14-35; 4:33; 5:12; 15:6-21;17:2,11; 18:28; 28:23; Rm. 15:19; 2Cor. 12:12, etc.)

Yet, besides comparison between Catholic and evangelical faith, the contrast between the church today and that of the book of Acts is one that calls for mourning and repentance, (James 4:1-12) and a corporate “dress code” of sackcloth and ashes till we find mercy and the Lord raises us up to walk toward perfection.