The thing of it is that whitetail deer are remarkably easy to kill, as long as you place the proper bullet in the right place. Anything from a 243 WIN up to a 45-70 will work. Some guys swear by the 223 Remington, but I think that's too small. For calibers, I like to stick to the middle of the road. To stay cheap, be flexible.
Look up the 300 Savage. I would call it the quintessential whitetail deer cartridge. Anything you pick should compare favorably to the 300 Savage. The 30-06 and the 270 Win both are more powerful than the 300 Savage. So is the 308 Win. A little overkill is fine. The 30-30 is not as powerful, but folks take a lot of deer with that cartridge. The 300 Savage is good on whitetail out to 200 yards, whereas the 30-30 (without a lot of help) is just a 150 yard cartridge. Look at the velocity and energy of the 300 Savage and figure that's your standard for taking a deer out to 200 yards.
If you really want to go cheap, a military surplus rifle is probably just fine. I have a couple Mosin Nagants in 7.62X54R that can be had for $70. They make nice deer rifles. Ammo is a bit of a problem, but I reload. I would shy away from the AK and SKS-- 7.62X39 is not a great deer round. Any of those 8mm Mausers you see in the C&R bin will work as well. Just make sure you have a source of ammo before you jump into it. However, if you're in a try-before-you-buy mode, $70-$130 for a reliable rifle is maybe worth considering. At $70, you can buy $70 worth of ammo and have af few years of fun before you decide to trade up. Remember that a military cartridge is generally meant to kill a human-size target. Deer are about has hard to kill as a human-- little different, but about the same.
Nowadays, you should be able to find a gun shop that will set you up with a used bolt-action or pump 30-06 or 270 for $250 to $300-- Or a Lever in 30-30 or 35 Rem. Your friends and relatives will probably have something in their closets that they will want to get rid of. My last two acquisitions were a rare Model 70 variant for $200 and a minty Remington 870 TB for $150. Both were closet queens that somebody from work thought I might like. The blue-book values are 2-4 times that. As long as you are flexible on calibers, you should be able to score something nice.
Use Gunbroker.com as a free source of current gun values. If someone offers you a $200 rifle and you go on Gunbroker and see the same thing selling for $800, you know you have something. Just remember that if you buy off Gunbroker you have to pay shipping and FFL transfer fees. I have done this frequently, but I knew what I was doing.
Lastly: remember that you don't need anything with "MAGNUM" in the name in order to kill a deer. The only difference for your purposes between a 30-06 and a 300 Win Mag is several pounds of additional recoil. The only exception I will make to this is the 44 Magnum-- as a pistol caliber it is a magnum. As a rifle round, it's about on a par with a 30-30-- no weakling, but nothing stellar.