Unless you're referring to some sort of over/under rifle/shotgun combination, I assume this is a either a shotgun or a rifle. In either case, the treatment is pretty much the same.
I would assume I should take it to a gunsmith for a standard cleaning and once-over?
Unless there is some heavy rust or deep corrosion, there is no reason you can't do the job pretty simply yourself.
You can buy a inexpensive cleaning kit in any gun shop or Wal-Mart, as well as the gun cleaner and oil. Ask any salesman for help. There is probably no need to completely disassemble the weapon unless it is in very bad condition and you can probably tell that by looking at it, trying the action and checking the barrel. Carefully clean the barrel and action of dust (after making sure it is unloaded, of course) and lightly oiling the barrel and working parts should do the trick nicely. Make sure you remove all obstructions from the barrel. Dry firing without ammunition in the chamber a few times while cycling the action will pretty much tell you if it is ready to fire. However, it would certainly be best to test fire a few actual rounds and you can also then check the accuracy of the sights.
BTW, what's a good, reliable brand of 30.06 shell?
Any brand you buy in the store is fine.
If it is a shotgun, #4 shot is good all around for small animals, such as rabbits and large birds. It can also be used for close in home defense. However, any of the "buckshot" calibres are better for defense, such as #1 and #00, as they will give a longer lethal range if that is a consideration. They will also kill larger game if necessary.