The only TiVos that have internal digital tuners are the Series 3 and the TiVo HD.
As for your TV, we have no idea what type of TV you have. In most (if not all) cases, even TVs that have digital tuners will need cable cards or a cable box to get the full digital cable lineup. (Without cards or a box, the best you can hope for with a digital TV is getting the over-the-air channels your cable system carries, since those aren't scrambled.)
If you want to get only one box and use it on the TiVo, and have that be your only source of digital for the TV, that's up to you. If you want the TV to be able to watch a different digital channel than what the TiVo is getting, you'd need a second one.
I wouldn't advise trying to use two cable boxes (of any kind) in the same room, tho'. They'd likely use the same remote codes and both boxes might respond when any button is pushed on the cable remote.