Lots of good advice from lots of peeps here, I'll throw mine in:
If you're on a very tight budget - and depending upon the terrain? - sight in your shotgun with slugs and stick to close shots. Depending on terrain.
Check out used rifles. Look over Gunbroker.com and Auctionarms.com, and various boards' classified.
I don't know if the 770 is crappy - never handled one - but apparently many dislike it. I know that Savage makes a very good rifle (I've had two). 30-06 is the 'gold standard', as someone said, but it's more than you need for deer, and it will - as several have said - boot you pretty good, esp. if you're a new shooter. Use hearing protection (lot of muzzle blast, which contributes to flinching, too). Now, if you've been shooting slugs from an 870, the 30-06 will seem relatively mild, except for noise/muzzle blast.
Lots of good choices out there. And listen up: DO NOT overlook a 2nd-hand Marlin 336 in 30-30. It has many virtues: it is pretty. It is all the power you need to as far as most people have any business shooting a deer at - a solid 175-200 yards rifle. It is classy. For ammo price.... it is the cheapest deer-appropriate cartridge out there. It isn't all that impressive when looking at a ballistics chart, but in the real world, shooting deer... it is will do as well.
I like the NEF handi-rifle w/heavy barrel; single shots appeal to me.
to go really cheap - get a low-end muzzleloader. I bought a CVA Buckhorn for about 80 bucks on post-season sale several years ago - there in the low $100-range now - it's a fine 100 yard solution. Killed a doe with it yesterday.