Bro,
I would stroll down to the local gun shop and chat with the folks down there and see what you can get for your budget. Each store has startet packages that will get you to the range. Get out and have fun! Even cheap sporting rifles have a fun factor. You WILL learn from experience and it is the very best teacher... NOT internet forums.
Consider a few "isms" here....
-Learn to master short range before frustrating yourself with the many technical challenges of long range shooting!!!!!!!
..."Precision shooting" is a sub-specialty of shooting in which specialized equipment is needed. A typical off-the-shelf rifle will shoot anywhere between 1/2 - 1-1/2 inches at 100 yards under ideal conditions, with a good shooter, and using good ammo. Double the size of that group for every hundred yards.
A "precision rifle" with proper hand-loaded ammo customized to the gun will shoot groups that are limited only by the ability of the shooter and his scope.
-The "art" of precision shooting is an amalgam of shooting skills, reloading skills, High Quality purpose-built equipment and PRACTICE. Wanting to start shooting "precison" is tantamount to wanting to learn how to drive in a Formula 1 car.
-NO ONE FACTORY RIFLE SHOOTS BETTER THAN ANOTHER! Brand loyalty must be recognized for what it is, and the arguments in favor of one brand over another are irrelevent. - More money DOES NOT mean more accurate.
-rimfire is a cheap, fun and practical way to get into the whole shooting expereince. If you shoot at a range, it will almost certainly put you in touch with people that are shooting centerfire rifles and you will get a feel for what is and is not practical.
-When you have a really good idea in your OWN mind, of the differences between a .308, a .300 Winchester Magnum and a .223, then you will be ready to start shopping for a ceterfire rifle that will fit your needs.
Every round you put down range adds to your skill and your experience. .22 is often all you really need.
oh... Cost of 223 vs. 22 ammo? - about 20X more per round