Your choices are autoloading rifles and, though both HK and Steyr are fine choices, they potential for affordable accessorization (should that be a consideration) is limited compared to the AR-15 variants, for which a great spectrum of aftermarket products are available.
If more precise shooting is also your intent, the autoloaders will provide very serviceable accuracy but I agree with the recommendation for a bolt action rifle for greater accuracy potential. The Remington Model 700 with the heavier "Varmint" contour is certainly an excellent choice, and there are others makes and models as well.
If you handload (or intend to) the .223 will provide you with more affordable practice and lower cost for components...more "bang for the buck".
You mention you intend to start hunting, but you didn't say whether your quarry is small game or medium game. That decision will affect others, as the .308 cartridge moves to the forefront of your two choices in either the autoloader or bolt action rifles.

You might consider a varminter class AR15. I have a Bushmaster Varminter... they have 24 inch heavy barrels and are capable of very high accuracy. I usually beat most Remington 700s at the range. I shoot quarter to half inch groups at 100 yards with it using hand loaded precision ammo. It is an outstanding varmint rifle and is lethal on antelope. It is on the light side for deer but is deadly out to 300 yards. I would not use it in heavy brush because of bullet deflection but in the open country it is borderlikne unsportsmanlike because it is so deadly. I will be using one for antelope in a couple of weeks (along with my 25-06). A 30-06 in Rem 700 is about as all purpose a rifle as you can get on the North American continent. I just prefer lighter bullets for most hunting in North America and therefore I use a 25-06. A .270 is another very good all around rifle. If you want a bolt rifle, the Rem 700 is a good choice as is the Tika T3.