Too many options, need help on a beginner Bench rifle

[Colorado Osprey]

I want to be able to send 10 rounds down range without it starting to walk. I shoot my savage and the first shot with a cold barrel is low left. The next 3 are on and then after that it starts to walk. I would like a rifle that will be on target first shot and can stay on target if I shoot 10 in a row.

By "fast" I mean 1-3 a minute. I have an AR for rapid.

That savage walking is a pretty easy fix. You have too much pressure on the barrel. Free float it and bed your recoil lug/action and it will settle down. Way cheaper fix than a new rifle.
500 yards is pretty easy for a Savage to shoot fairly accurately.. maybe even better than you can presently shoot.
Are there better rifles, sure.... but I'm not sure you need one to shoot 500 yards. My son is currently shooting a factory model 111 in a .270 WIN with a slight stock upgrade (Boyds $100) and can easily hit 6" steel plates at 400 yards with a Leupold 3-9x40 Vari X II. He is 13 years old.