Given the criteria of medium North American game, light recoil, reasonable meat damage, affordable, and available my top choices are simple.
308, while I usually prefer the extra speed of my 30-06 the 308 does kick considerably less and is a suitable cartridge for any medium game, just a great cartridge. 150gr SSTs, Speer BTSPs, or SGKs make quick work of any whitetail deer.
7mm-08 If there is anything that can beat the versatile 308 it is it's necked down cousin the 7mm-08, noticeably lighter recoil, better downrange ballistics and all the efficiency you love about it's parent case. While it is easy to compare to the 308 I actually compare it more with the 270 Win, I am pushing 140gr bullets to 2950fps burning about 10 less gr of powder then the 270 at the same speed reducing recoil and muzzle blast considerably, and doing it all in a light short action.
6.5x55, 260 and 6.5 Creedmore. I lump these all together since ballisticly there is no real difference between them. If getting some range is your game the downrange ballistics of the 6.5mms is legendary. While the 308 CAN launch .600+ BC bullets it can only manage around 2400fps while my 6.5x55 can top 2,800fps with them making very high BC bullets practical for hunting. While the 6.5mm is the smallest caliber on my A list they are still VERY effective on game, in fact none of my magnums and 30 calibers have ever done the kind of damage to the vitals that my little 6.5 has. Recoil is very light and meat damage is less then the 243 I used to use.
30-06, OK so it is not a light kicking rifle, but if you can learn to handle a little recoil (they are not THAT bad, much less then magnum slugs) this is probably the most versatile cartridge in existence from yote control to African plains game and everything in between. Unlike the 308 the 06 is comfortable with 200 and 220gr bullets should you end up hunting larger game in the future.
For rifles I like bolt actions and single shots, sure leavers and semis are faster but all I need is one good shot and it is game over, I have never missed. My favorite rifles are usually Savages, Brownings, and my fantastic Tikka T3. Ruger and Winchester make great Mauser style actions which I prefer but I have never been able to get them to shoot quite as tight, though their accuracy is more then acceptable for medium game hunting, so pick what feels the best to you.
Favorite optics are from Leupold and Nikon, sure there are higher dollar optics out there but in my book Leupold still has the best glass and Nikon is without question the best value.