What type of woods? How close are your shots likely to be?
A tight woods gun might be served well with a Marlin 336 in .30-30 or .35 Remington, where shots are likely to be less than 150 yards and more likely to be inside of 100 yds.
A rifle for taking open field shots out to 200 yds and beyond, A .30-06, .270 or 7mm should be fine. Savage, Tikka, or used rifles like Winchester model 70 or Remington model 700 would be good choices. The Savage 110 can often be found used, with a scope well inside your budget, and they are surprisingly accurate for what they cost.