Look at...
Savage 11's or 10's...I have a Model 11 in 7mm-08 and a Model 10 in .308...they both shoot great and can be had for under $500...get one with an accu-trigger...
Tikka T3...have one in a 270WSM...shot sub-MOA out of the box with 3 different handloads...adjustable trigger breaks like glass, bolt as smooth as butter and detachable mag...well worth every penny I spent on it...
Browning A Bolt (or X Bolt)...I have an A bolt in 25 WSSM and it too shoots very nice...adjustable trigger, detachable mag...shoots very well...have heard the same about the X Bolts...but may be over your $500 budget...
Weatherby Vanguard (AKA Howa 1500)...my daughter and niece both shoot Vanguards in .243 and they too are very accurate and handle nicely...
Those are my recommendations...many others will add several rifles to the list (Remington 700...I love the older ones not so much a fan of newer ones) and so one and so on...bottom line is find a rifle that has the options/accessories you want that fits you and feels good in your hands...fit and comfort are the most important elements when I buy a rifles these days.
Good luck in your search.