Obviously, this is just my experience and everyone will have different perspectives, plans, hopes and interpretations but hopefully this will give you a bit of an idea of what you're in for!
As you get close to landing, the airline crew will give you a landing card to fill out. It has questions about your passport information, where and for how long you are staying in Ethiopia, and what the purpose of your trip is. Try and fill them out before you land (it is nice to have a pen on hand for this) so you don't have to try and write while standing in line. If you don't know the address and phone number of where you're staying, don't sweat it - just write the name of the hotel or guest house.
We flew Ethiopian Air from Dulles to Addis. Had I to do it over again, I would take a different airline for two reasons - one, Ethiopian is not exactly luxurious. The tv was broken most of the time, the seats were even more cramped than most flights. Two, it is TOO long in a small space without being able to get out. I would opt for a stopover somewhere in Europe, even if it meant having to stay a few extra days in Addis for a transit visa. If you have the money and are flying Ethiopian, upgrading to 1st class is $700 one way - something we seriously considered (although didn't do) for at least one of us.
Put your landing card with your passport and your $20 (the fee for the visa, have a new $20 bill if possible) somewhere you will be able to easily access it once you're off the plane. If there is more than one person traveling make sure each person has a landing card filled out, their passport and a $20 bill.
The international terminal the Addis airport is the nicest airport I've been to in Africa, barring perhaps Johannesburg. It is modern and relatively easy to navigate. You will deplane and follow your fellow travelers down a flight of stairs or down the escalator. When you get downstairs you will be in the Arrivals Hall. There is a large open area with customs booths at the end. This is where people who already have visas should queue. If you are getting your visa upon arrival (as we did), you go a little to the left, into a line that leads to door marked "visa". The line leads in to a room where there are 4 or 5 desks, each with a person sitting there processing visas.
When it's your turn (and we didn't wait in line more than 20 minutes) you just hand them your passport, arrival card and money and they fill out some papers, take your money and hand you (1) your passport with your visa and (2) a receipt for your $20. Put the receipt in your passport, go out the door (the only way you can go) and get in line in the large, open arrivals hall. There are many lines each leading to an individual booth, you can get in any one of them except the one marked "diplomat" (unless you are a diplomat, in which case cool!)
Wait again (shocking, I know) and when it's your turn approach the customs booth. Give them your passport, answer any questions they have (they didn't ask us anything), and they will stamp your passport and send you on your way.
Once you exit customs you will be in the baggage claim area. On the wall in baggage claim closest to the customs area is a bank. (The bank actually has a window in the arrivals hall too, we chose to go ahead and get in line and wait to change money until we were done with all the lines.) Our agency told us only to change $100 the airport because the exchange rate was not as good there, but I wish we had changed more. The difference seems to be less than a penny to the dollar, and it was a pain to have to do it later.
To change money, just give the person at the bank your passport and the amount of money you want to change. They will write you a receipt, have you sign it, and then give you your bieer. Also in the baggage claim area are luggage carts. They are free and, if you brought as much crap as we did, wonderfully useful. We got one, brought it over to the conveyor belt, and waited a long time for our bags. It was a bit of a mob scene- lots of people pushing to get their things, so just take a deep breath and if it's really stressing you out maybe step back a little until the crowd has cleared. It also helps if your bag has some kind of easy to stop mark – my dad makes big "x"s with duct take on each side of the bag. Not so pretty but gets the job done. There are men who work at the airport who wear brown coveralls. If you want you can give them your luggage check tags and they will get your bags for you. I saw a lot of Ethiopians doing this, but don't know how much they paid them for the help, although not much, would be my guess.
So, you have your visa, passport, bieer, and luggage. You are ready to go. If you have nothing to declare (and you would know if you did), you will head for the exit.
Before you leave stop for a moment. If you are with someone, take their hand. You're about to see Ethiopia! Try and let all the stress wash away for a moment, and appreciate where you are and what you are about to see and do.
Then it's out into the madness. As you walk to wards the glass sliding doors to exit you will see a sea of faces. People waiting for people, people who want to give you a ride, recommend their hotel, help you with your luggage, etc. Don't stress. Just walk like you know what you're doing (fake it till you make it if need be) and say a polite but firm "no" to anyone whose help you don't want. Find whoever is picking you up (or, more likely, they will find you) and they'll take care of you, and you're off!