My own first was a 30.06, so I have a special affinity for it, but I honestly don't think you could go wrong with .243, considering the trajectory performance, low recoil, economy, and availability. Not to mention the fact that it'll be much easier to find a good used rifle in .243, since it's a popular choice for first time hunters anyway.

I've heard a few people complaining about it's performance at long range, but compare it to the heavier rounds in this chart and you'll see it's not really an issue http://tinyurl.com/ammo-ballistics-chart

At any rate, I hope you find the one that's right for you and your's! Good luck!