You have to get the right tool for the job. Hunting grouse with a centerfire rifle is like driving finish nails with a 6lb sledge hammer... the sledge will sure do the job, but you're not going to like the finished project. There is a BIG difference between a deer and a small upland game bird, and the right tool for the job is different in both cases.

If you simply can't afford but one gun, I'd have to agree that a 12 gauge pump shotgun is the best choice. It's the multi-tool of the gun world and with the right combination of shell, barrel and sights can be used to humanely harvest anything that breaths air, within the limits inherent of the weapon.

Another option you might consider is to get yourself a Thompson/Center Encore and get both a 12 gauge barrel and a centerfire rifle barrel in your choice of caliber. Since you're asking this question I'd guess that you're not an experienced rifleman, so I'd STRONGLY recommend that you start with a lighter recoiling cartridge as your first deer rifle. I'd recommend you'd consider the .243 Winchester, .25-06 Rem or at most the 7mm-08 Rem. All are excellent deer cartridges that won't break the bank your bruise your shoulder.

Mike