What are your recommended alternatives to safari in the Serengeti?

In my opinion there's not much that tops Serengeti National Park. As far as the wildlife and scenery goes there's nowhere better for a safari in Tanzania.

The only downsides I can think of are that visiting the Serengeti can involve lots of driving (to see the migration river crossings takes two or three days driving), also it can be expensive and, at peak times, busy (although far quieter than neighbouring Masai Mara in Kenya!)

For those who prefer less driving, I'd recommend Ngorongoro Crater as a good runner up to the Serengeti. Ngorongoro's compact size (the crater is only 20 kilometres across) means you can see more with much less driving. All the big five can be seen here and as a bonus you have have a better chance of seeing rhinos here than in Serengeti.

If you want to escape all crowds and get off the beaten track, I highly recommend Ruaha National Park. Located in the midwest of the country, it surprises with outstretched savannahs and plenty of animals, together with some amazing lodges. The downside (or upside, depending on what you're looking for) is the distance – it’s pretty far from everywhere.

And finally if you're looking for a lower cost safari in Tanzania you could book flights to Dar Es Salaam, and from there do a safari in Mikumi National Park. Relatively small, it still offers open savannah with all your usual safari suspects except for rhinos, along with the most beautiful sunsets and sunrises.