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Tikka T3 Varmint in 308, new for $1200(ish), decent scope new $1000(ish), rail and rings--good quality! $250(ish) and you are right on budget with a much better rifle than a Remington.
Definitely look on the EE. You can find a pretty good package on there for your budget. For my entry level gun I went with a Savage fcp-k in 223. Also comes in 308. Has an adjustable trigger and accustock. It shoots lights out. Ruger precision rifle is also an option. Try to pick something g with the least amount of upgrading to start with (ie adjustable trigger decent stock etc.). For scopes I was looking for something with a good warranty like a vortex viper PST. However also looking at sightron siii.