Tipping is customary in Tanzania, and hardly anyone explains it, so people either worry too much or tip too little. So here's how it works. These are the usual ranges, not rules, and nothing is compulsory.
Your safari guide
The person who makes the biggest difference to your trip. The usual tip is somewhere around 15 to 20 US dollars per vehicle per day. That's per vehicle, not per person, so a family splits it. Give it at the end of the trip, not every day.
Camp and lodge staff
Most camps have a shared tip box that's split among everyone, including the people you never see. Around 10 to 15 dollars per guest per day is usual. Putting your tip in the box instead of handing it to individuals is the fairer way, and it's what camps prefer.
On Kilimanjaro
This one's different, and bigger, because so is the crew. A climb involves guides, a cook and a number of porters, and the total tip pool for a week-long climb is commonly several hundred dollars for a small group, shared out by role. Ask your operator for the exact breakdown for your crew size before you go, and bring it in cash. You could easily argue this is the most important money you'll spend on the whole trip.
The practical bits
Bring US dollars in cash, in decent condition. Notes printed before 2009 are widely refused in Tanzania. Smaller notes are more useful than one big one. There are ATMs in Arusha and Moshi but not in the parks, so take out what you need before you leave town.
If the service wasn't good
Then tip less, and please tell us. A tip is a thank-you, not a bill, and we'd much rather hear that something went wrong than have you quietly pay for it.