While Paul eventually would be called and chosen later on in Acts, and he would at that point likewise begin to preach and teach as an apostle of God the Father and the Lord Jesus Christ, I think to best answer the original question regarding exactly what the Apostles Doctrine was, as of Acts 2:42, it is important to stick with the Twelve before moving on to anything Paul said or wrote.
From Acts 1:1-4 (KJV), we know the following:
1 The former treatise have I made, O Theophilus, of all that Jesus began both to do and teach,
2 Until the day in which he was taken up, after that he through the Holy Ghost had given commandments unto the apostles whom he had chosen:
3 To whom also he shewed himself alive after his passion by many infallible proofs, being seen of them forty days, and speaking of the things pertaining to the kingdom of God:
4 And, being assembled together with them, commanded them that they should not depart from Jerusalem, but wait for the promise of the Father, which, saith he, ye have heard of me.
Here, we see that Jesus through the Holy Spirit gave His apostles commandments. We also see that prior to those commands being given in and through the Holy Spirit, Jesus spent forty days with them speaking with them things pertaining to the Kingdom of God.
In the Gospels, Jesus frequently spoke to them the same things, so I suspect the things Jesus said to them during that forty day timeframe were reminders and summaries of the things He had already said over the course of His pre-Calvary ministry.
We can therefore see that the doctrine the Apostles preached and taught was not really theirs, per se, but rather, was the doctrine taught to them by the Lord, i.e. the things pertaining to the Kingdom of God (See also, e.g., Acts 13:12).
Interestingly, in John 7:16, Jesus said "My doctrine is not mine, but his that sent me."
This shows that, according to Christ, the doctrine He taught and espoused wasn't really even His, but rather, belonged to and came from the Father, i.e. God Himself.
This makes sense in light of John 13:20,
Verily, verily, I say unto you, He that receiveth whomsoever I send receiveth me; and he that receiveth me receiveth him that sent me.
There is an unbroken doctrinal line descending first from God the Father, to His Son, the Lord Jesus of Nazareth, and then, to the Twelve, then eventually to Paul of Tarsus and the other apostles and elders of the 1st Century Church.
As regards specifics, I recommend looking at Hebrews 6:1-3,
1 Therefore leaving the principles of the doctrine of Christ, let us go on unto perfection; not laying again the foundation of repentance from dead works, and of faith toward God,
2 Of the doctrine of baptisms, and of laying on of hands, and of resurrection of the dead, and of eternal judgment.
3 And this will we do, if God permit.
The above passage shows what the principles of the doctrine of Christ are, that is, what the original foundation that was laid, was:
- Repentance from dead works
- Faith toward God
- Laying on of hands
- Resurrection from the dead
- Eternal judgment
- Going on to perfection
These seven things seem to encapsulate the core teachings of the apostles.