Hi Dan,
Just back from Tanzania - lucky you that you are going, it's a wonderful place!
Starlagurl has given some great links for accommodation - budget really depends on what you want - there is a wide range on tripadvisor given below, but here is another link just in case:
http://www.hostelworld.com/findabed.php/Ch...ountry.TanzaniaAlso Starlagurl is right about the food, it depends again what you want - I would say that you could happily eat and drink for between 5 and 10 US Dollars a day if you get food from street vendors. Restaurants of course cost more.
The warm sleeping bag will depend upon where you are from - I was in Arusha a couple of days back and it was around 30 C, although it does get cooler at night - more like 16 C at night - for me, being from England, this is very warm and a sleeping bag would be too hot, so I guess it depends what temperatures you are used to.
Have a great, great time.