[T. O'Heir]

"...my budget...I can buy used..." Smart man who thinks about his budget first. Buying used is not a bad idea either. Takes a great deal of abuse to damage a modern hunting rifle.
You might find a decent used rifle, with a good scope, for not a big pile of money, in your local gun shop. Not the best time of year though. Best buys are usually just after deer season. However, you might get lucky and find one that the retailer still has 'em and will/might give you a better price.
Gun shows are another place to look. Know retail prices when shopping there.
Savage package rifles are good rifles. The scope is serviceable, but low end scope.
Don't limit yourself to a .243 either. Any of the mid calibre cartridges(.24, .25 or .26), that aren't hard to find ammo for, locally, will do nicely.
A .30-30 lever action is an 'I want one' thing. Felt recoil is more for the power of the cartridge(out of a Win M94 anyway. Not so much with a Marlin) and they are not as accurate as any bolt action.
"...DO see the .243 being thrown off target much easier than the 30-30..." Nope. A 170 will get tossed hitting a twig just as much as a 105 will.