There are two options to plan your Tanzania safari on the Northern Circuit. You can do it ahead of time, finding out a safari company through guidebooks, recommendations, and the internet, or you can just turn up at Arusha and look around there. Each of these techniques has advantages and drawbacks. Planning ahead makes it more likely that you'll get bookings at the Tanzania safari lodges you want (they fill up during peak season), but can be scary, since it involves a certain leap of faith in the safari company's honesty and competence. Showing up in Arusha city and looking allows you to check out the tour company more thoroughly, but it may take a while. The city itself is unpleasantly touristy, and you'll have to sort through a lot of competing offers of dubious quality.
Start by deciding what kind of Tanzania safari tours you want ; make a list of the safari parks you want to visit; and decided on staying in safari lodges or Camps.
You may want and need safari lodges for two reasons; they are more comfortable beds and air conditioning. The beds are fine, but in early August, air conditioning turns out to unnecessary; from that point of view, camping safari will not be a problem. Otherwise, the decision rests on your Tanzania safari budget and your preference in ambience. The price difference between the two is about $40 per person per day, $130 in lodges versus $100 camping. The lodges are, for the most part, large, luxury, or semi-luxury hotels, with grand buffet meals in cavernous restaurants and fleets of staff to carry your luggage for you. The campsites are campsites, with very basic plumbing facilities and no fences around them, so that, in principle, lions can wander into your camp at night.
Whether you choose lodges or camping safari in Tanzania, food is provided. Vegetarian food is no problem in the lodges. Kosher food should be OK too, depending on how strict you are; you can go veggie if you want to be on the safe side.
After you have decided what you need, choosing a safari company turns out to be surprisingly easy. Check guide books for listings of Tanzania safari companies in the low- to mid-range budget, and check out their websites. Email the companies with descriptions of what you need, and decide on a custom made safari itinerary.
Whatever you choose, you'll absolutely have to have a 4x4 Safari Land Cruiser and a guide to enter all the Tanzania National Parks including Ngorongoro and Serengeti National Parks. Prices may be negotiable, but only up to a point, as there's a large park entry fee (up to $71 per person per day).
So that's the practical order of how you choose your Tanzania safari tours. But the other thing that matters most, the thing that provides you with the questions to ask your safari company, is the choice of Tanzania safari itinerary. There are a number of parks in the northern of Tanzania, each with a different character. In the course of a 5 day Tanzania safari, you can go to Tarangire, Serengeti National Park, Ngorongoro Crater, and Lake Manyara; this is a pretty intense travel schedule, as the distances between the parks take a while to cover. Most Tanzania tour companies will suggest that you do something similar, a sort of tasting menu of the parks, but if you want a more leisurely pace, it's perfectly possible, and you can choose to have longer stopovers, which would allow you to spend one morning looking at the vista in a lodge or several hours watching the same group of animals (lions hunting, for example).
Each national park is unique. Tarangire is a plains ecosystem around a permanent river, and is supposed to be good for elephants. The Serengeti park is a vast, vast plain teeming with pretty much every kind of wildlife. To get there from Tarangire takes most of the day, as the road passes through Ngorongoro Park, so it's important to spend two nights in Serengeti if you want to make the trip worthwhile. Ngorongoro Crater is spectacularly beautiful and has one of the densest populations of animals; it's also the only park in which it's possible to see black rhinoceros. It's also the most touristy of the parks; if you're bothered by seeing other safari vehicles, make your stay there a shorter one. Lake Manyara is known for its tree-climbing lions and one of the best bird-watching spots to be seen. There's also Arusha National Park, which is less touristy than most.