Hi - a tour company suggested that I start my trip off hiking Kilimanjaro first before the safari so any fitness I have I don't lose. However, I have read that I should safari first so that I recover from the travel: rest and hydrate. What has been people's experiences?

From my experience, the vast majority of people climb Kili first and then go on safari. Reasons vary, but I've never heard anyone say afterwards that they wish they had done the reverse. At the same rate, a few have done the safari first and they didn't regret that decision either.

A few considerations:

1) Jet lag. If you are traveling from many time zones away then jet lag is a consideration upon arrival. For me, coming from the US, it means that I often wake up somewhere around 4am and can't go back to sleep. I'm not sure how this would impact a Kili climb...

2) Most seem to climb Kili right away to 'get it out of the way' so they can enjoy the rest of their visit.

3) Acclimatization: Some will try to sleep a few nights at a higher altitude (maybe at the crater?) but little tells me this really helps that much.

The good thing is that I haven't heard of anyone who has ever really come to regret the decision regarding which to do first.

mdtravelhealth.com good info in addition to above altitude illness issues climbing Kili consider a preventive like Diamox. Most elect Kili first very strenuous then chill
on safari what I did years back.

Kili first, then safari is the usual for reasons above.
If traveling a long distance, or even if not, suggest you build-in a day on arrival to avoid jet lag or if delayed baggage... then climb on Day 3. Do include day for acclimitization on the mountain. Good luck!

Thanks everyone. You've been helpful. I think I will build an extra day on arrival to catch up : )