A week is a good amount of time. As mentioned by the above poster, one or two days in the Ngorongoro Crater is typically enough. With good communication between guides, you should be able to see a lot given that most animals are in the open field, and confined to a relatively small area (the crater). Also Ngorongoro vehicle entry costs are expensive, so you may have to pay more if spending more than a day here. The Serengeti is a different story - it's huge, and the animals are much more spread out and harder to locate. Therefore you'll need many more (3-5) days to see this area and be satisfied. I think you'll find this environment much more natural. You often get traffic jams at Ngorongoro when a dozen land rovers line up around an animal. This doesn't happen as much in the expanse of the Serengeti.
The Serengeti is about 4 hours from Ngorongoro, on a bumpy dirt/gravel road. It's common to break up that drive with a stop at Olduvai Gorge. You may want to consider visitng Lake Manyara and/or Tarangire before Ngorongoro/Serengeti. Another thing to consider to break up the time in the vehicle is a trek on the Ngorongoro highlands with a Masai guide.
The safari cost mostly depends on your accomodations. About $200 per day for camping and $500 per day for lodging for a solo traveller, which should include everything (no more out of pocket costs).