To put in my 2 cents:
I was talking about a real competition or sniper rifle, if you just want an accurate bolt action rifle, go for the Remington 700p and plan to upgrade later. They are great platforms and perform very well out of the box. You could also try the Savage Arms Police rifles. Also, you are going to want a decent scope which will cost around 400+ depending on if you are wanting to compete or "Snipe" as well as the range you are working in.
Caliber choices can be extremely difficult. Once again, it all depends on what you are going to do. Most competitions rifles use .308 for its accuracy and cost. The .308 should work out to around 900-1000 meters. If you are going farther, the .338 lapua is an amazing cartridge. A random cartridge (which happens to be my favorite, but extremely hard to find in bolt action) is 6.5 grendel. It is a little bit smaller and lighter than the .308, but it flies faster, hits harder, and is effective out to about 900-1200 meters. The .50 BMG IMHO is a very useless round for civilians for many reasons:
1. At $5 a round, it is extremely expensive.
2. Rifles cost considerably higher and weigh more than twice as much.
3. Is a comparitively inefficient round for "sniping"
4. You get restrictions when using a round that big both at gun ranges and just plain owning.
In general, we could argue calibers until the cow comes home. It all depends on how you will be using the rifle.
Semi-auto wise, I agree on the AR-10. It is very accurate, but not as reliable as some. Another great choice would be a Springfield M1A. They are extremely accurate and extremely reliable.
As for the Mosin, I would suggest that you don't waste your money unless you collect. Mosins are very innacurate (all of the ones I have seen and also based on how they were built). And it having a big kick is an unbelievable understatement. They say that the first shot throws you shoulder out of socket and the second throws it back in
. With the recoil, you need to be careful that you don't start out too high because you could develope a flinch.
In order for us to answer more specifically, you need to give us more info.
1. Are you a new shooter with RS? What is your experience?
2. What is your budget?
3. What are the ranges you will be shooting at?
4. What kind of competitions do you want to shoot in.