Most of this is simpler than people expect. The one thing worth getting right is arriving early enough.
Kilimanjaro International, not Nairobi
Kilimanjaro International Airport, code JRO, sits between Moshi and Arusha and is about an hour from either. It's small, the arrivals hall is one room, and it's the right airport for a climb. Dar es Salaam is a long way south and adds a domestic flight. Nairobi works as a connection but adds a border crossing or a second flight, and it's only worth it if the fare difference is large.
The visa, in outline
Most visitors need one, there's an online application and there's ordinarily a visa-on-arrival option at JRO. The details, the fees and the eligible nationalities are set by the Tanzanian government and they change, so we describe the shape and point you at the source: the official immigration portal. Apply online in good time if you'd rather not queue at midnight after a long flight. There's more in the visa article.
Moshi or Arusha?
Moshi is closer to the mountain, smaller, quieter, and on a clear morning Kilimanjaro is simply there above the town. Arusha is bigger and busier, has more choice of hotel, and is the right base if you're adding a northern safari afterwards. Transfer times to the gates differ by less than an hour between them, so pick on the rest of your trip rather than on the driving.
Arrive a day early. At least one.
You want a full day before the gate for the kit check, the briefing, hiring anything you're missing and sleeping off the flight. If your bag doesn't arrive with you, that spare day is the difference between hiring replacement gear calmly and starting a seven-day climb in the wrong boots. Two days is better if you've crossed a lot of time zones.
And a day afterwards
Don't book a flight home for the evening you come off the mountain. You'll be filthy, euphoric and extremely tired, and the descent sometimes runs late. A night in a hotel with a shower and a bed is part of the trip.
Money on arrival
There are ATMs in Moshi and Arusha and a bureau at the airport. Bring US dollars in cash for tips, in notes printed in 2009 or later, because older ones are widely refused. Cards work in hotels and bigger restaurants and nowhere on the mountain.
If you're adding a safari
Climb first, safari afterwards, nearly always. You'll be fitter and fresher at the start, and spending the days after the mountain in a vehicle with a cold drink is a much better recovery than doing it the other way round. It also means a safari delay can't eat into your climb. Arusha is the better base for that version of the trip.