If that $2000 includes glass, my opinion would be to look at a Savage rifle.
Something like this Savage Arms Firearms > 10 FLCP-K
Would give you an excellent shooter. Take the money you save and get better glass out front. If you decide to upgrade, the Savage action is the AR of bolt rifles and you can have a completely upgrade and reconfigure with a few tools in a few minutes.
Not intending to start a raging debate with this one but I would say that the difference in accuracy between a Savage and a Remington straight from the factory is negligible to say the least. Both can be accurized and both are proven shooters that will suit your needs. If you're looking for a setup with the best performance to value out of a factory rifle I'd say go money heavy on the glass side. The performance you will gain (if any) spending extra on the factory rifle isn't worth it compared to the performance gain to be had taking that money you would have spent and investing in a better scope. Let the shelling commence!
Just my .02. Best of luck and welcome to the club!