it seems that the trend is towards semi-autos. if that is the case for you, i would suggest a springfield m1a. it will cost a bit to get one, but if you don't start with something good, you probably won't like it. .308 ammunition is cheap and good for practice. learn to reload in order to get the maximum potential from your rifle and your ammunition. if you want a real challenge, pick up an m1 garand somewhere and restore it to pristine condition while learning to use it well. both the garand and the m1a are known as excellent shooters. from what i remember, you can get either an m1a or a garand for under a thousand dollars if you shop carefully. if you find a good used m1a, you could save a considerable amount of money.
if you are in the market for a bolt action, the first i would suggest is a remington m700adl in .308 or .30-06. if you plan to shoot more than hunt, get the .308; if you plan to hunt more than shoot, get the .30-06. both are outstanding cartridges, but i believe that the .308 has a slight edge when target shooting, while i also believe that the .30-06 has a slight edge in hunting. the m700 is an accurate rifle to begin with and with a little tuning can be extremely accurate. another thing about the remington m700 is that they are the "novas" and "mustangs" of the gun world. parts, both standard and after-market, abound. any gunsmith in the world can work on them, and customization and accurizing is easy.
savages are also very accurate, but the craftsmanship leaves a lot to be desired, in my opinion. the howa m1500 is a lower-cost option with fairly good craftsmanship, and is also a good shooter. i would certainly get it before i got a savage, and they are about the same price. once again, get the best that you can afford, or you may lose interest quickly. there are other options: sako, tikka, browning, and winchester to name a few. these all have push-feed actions, as opposed to mauser actions such as cz, charles daly, ruger et al. i very much prefer mauser actions for hunting, but the conventional wisdom is that they are not as good for target shooting. with that in mind, my recommendations would be a push-feed, and specifically the remington. all of my centerfire rifles are mauser-style except a lever-action .30-30.
i probably don't have to tell you this, but it would also be wise to invest in a military-style sling. i haven't a bloody clue how to use one, but i do know that serious target shooters rely on them for stability in a wide variety of situations.
once you choose the rifle, it's time to start looking at scopes!