Jamiet,
Thank you for the question. I appreciate people trying to understand the Christian belief system, rather than insisting on imposing misunderstandings.
Different denominations and theological systems do answer this question differently, so any "specific list" you find will probably differ from other specific lists. Here is one example (with which I do not personally agree in every particular):
http://www.reformed.org/documents/wcf_with_proofs/ch_XIX.html
This one basically says that only the ten commandments still apply. It is held by most forms of Presbyterianism, among other reformed churches.
Anyway, to answer with some generality: before Jesus died to sacrifice for our sins, God could/would not indwell sinful people through the Holy Spirit in the way that He now can/does (since God cannot stand sin). In this time of history (post-Abraham), His relationship with the human race was through one special chosen people, the Jews; and He gave them laws (via Moses) for several distinct reasons. First, due to their not having the Holy Spirit indwelling them, external laws were more necessary to tell them exactly how to behave. Second, He wanted them to be set apart. It was deliberate that they should be visibly different from every other people, in behavior and customs, and sin was particularly intolerable for this reason, lest it spread. Hence, the punishments were, by today's standards, very severe in a lot of cases. In addition to maintaining the purity of the people, this served as an image to the world of how seriously God views sin. It is no longer our place (fortunately, to my taste) to execute (part of) God's judgments in the New Covenant, as it was in the Old; but God's commands were always expressions of His justice. Sin is no less abhorrent to Him now, and will not ultimately be less punished.
With the coming of Christ and the dawn of the New Covenant, God's relationship with man underwent quite a profound change. People of any race and location can now be God's people, and it is their behavior through the Spirit of God, not their identity as a separate nation, that is now the visible body of God's people on earth. Thus, there is no longer any need for a theocratic state to impose uniformity and morals through the arm of force.
One important thing to realize (which many Christians, even, do not) is that Israel of the old covenant was God's covenant people; thus, the correct modern analogy is to the church, not to the modern nation (i.e., not to America, the UK, or whatever). Thus, the correct analogy of the old laws to the new context would be to the church punishing Christians who have sinned, NOT to America punishing unbelievers who have sinned. As the covenant people (church) is now spread throughout all nations, and is not associated with any government, one can see that it would be impractical for it to carry out executions, even if it weren't also wrong (which it is, since God has removed that command, and there is no right to kill absent such a command).
There are many reasons still to study the Old Testament, even though the Mosaic laws no longer apply. For example, many of its prophecies still apply to our current time or to the future; it helps us understand the arc of God's relationship with man (as FlemGem said, the Bible is a story, and you'd be cutting off the crucial first two thirds if you discarded the OT). It is also crucially important for the proper understanding of the NT. The OT is much more than just a list of rules, and while those no longer apply, there are a lot of promises made that form the context for the New Testament, and are necessary background for understanding it. It also helps us understand the character and nature of God. For example, even though we are no longer commanded or even allowed to kill adulterers, the fact that God did order that at one point suggests how seriously He views sin. That in turn helps make sense of the New Testament.
I'll close with the promise, in the Old Testament book of Jeremiah, of the coming of the Holy Spirit and the New Covenant:
“Behold, the days are coming, declares the Lord, when I will make a new covenant with the house of Israel and the house of Judah, not like the covenant that I made with their fathers on the day when I took them by the hand to bring them out of the land of Egypt, my covenant that they broke, though I was their husband, declares the Lord. For this is the covenant that I will make with the house of Israel after those days, declares the Lord: I will put my law within them, and I will write it on their hearts. And I will be their God, and they shall be my people. And no longer shall each one teach his neighbor and each his brother, saying, ‘Know the Lord,’ for they shall all know me, from the least of them to the greatest, declares the Lord. For I will forgive their iniquity, and I will remember their sin no more.”
Anyway, I hope this has been helpful at all.