There are so many good choices out there. Between the 308 and 3006 you have about two hundred feet per second spread. They will both shoot bullets from 110gr to 200gr. The best bullet for deer is in the 165 gr range or lower. For elk I would go with 180 or bigger.
Both rounds have been used by the Military. The 3006 was dropped for the 308 because of weight and very little was given up in F.P.S.. In short you could carry more 308 than 3006 carrying the same weight. The military still uses the 308 in sniper rifles. They don't use the 3006 for anything. It is also cheaper to load the 308 than the 3006. You use less powder in the 308.
As for the rifle there is a Co called Thompson Center. They make a bolt rifle that goes for $500 that will give Savage and Remington a run for their money. They guarantee 1"MOA three shot group. That is more than you need for hunting. I have a Remington 700, Model 7, and Savage model 10. They are good rifles I have no complaints with any of them. They are all older rifles and they all cost more than $500. I don't have a Thompson bolt gun yet. I do have two of the single shots that the sell and they are great. I have the Encore with two barrels, one in 243 Winchester and the other in 358 Winchester. Both rounds are made from the 308 case. I also have the Contender with a bunch of barrels for it. The one that I use the most is the 223. It is cheap to shoot and very accurate.
If it was me I would go with the 358 Winchester. Load it with 200 gr bullets and go hunting. The only problem is I don't know of anyone that makes the 358 Winchester in a bolt rifle any longer. You could find one used, then they are not cheap. The last one I saw on Gun broker was going for over $800 and it didn't have glass with it. It is a good round. It has taken just about every thing in America. I have never read of anyone taking the big bears with it. Most people that hunt the big bears use the 35 Whelen or something bigger.
I hope I gave you some information that was of use. Good luck with what ever you decide and I hope you have a safe hunt.
For hunting I would try to go with a 50mm+ objective just for the light gathering. Other than that I would look for quality and not be cheap, good glass can be the difference between a great hunt and an unsuccessful one. Burris, Leupold, Nikon etc, all good...just avoid the really cheap glass out there.
https://www.indianagunowners.com/fo...0-featherweights-five-calibers-available.html
He has either .270 or .30-06. I think you need to slide your budget up another $550. One of these rifles plus Vortex glass from A&A Optics. You'd be happy, happy, happy.