My first big recommendation is read the Lonely Planet and keep it for useful information such as language tips, money stuff, and maps, and then just ignore all its recommendations. Below are the city specific recs, and at the bottom some more general advice.
We arrived in Yangon (Rangoon) with only two nights booked in a hotel and I think that was a good decision. On out first day we sought out a travel agent (Good News Travel - they have a great website) and with their help booked ourselves through the next two weeks. Now, we were traveling in the low season and I have heard that this is not as easy in the high season. If you know you want to do Bagan and Inlay, then best to get those flights and hotels booked before you go. The only town we had a bit of difficulty finding a place to lay our heads was Hpa'an and I suspect that has more to do with a dearth of available places and not overcrowding.
We all really liked Yangon, it is for lack of a better comparison, the New York of Burma. It's a mess too - traffic is terrible in part because motos and scooters are banned, a very unusual thing in this part of the world. This is where the business community is growing and the diplomatic community has remained though the capital has moved north. The hotels are the best and they are all extraordinarily expensive. Like San Francisco during Apple Week expensive.
Go to the Shwedagon Paya, go to the Bogyoke Market (Bo-j-oke), discover Chinatown, and watch the city changing as you spend time there. We had wonderful drinks at the Governor's Residence (an Orient Express hotel), and really liked our stay at The Savoy. For a cheaper alternative to both I also recommend The Classique Inn and the Alamanda Hotel.
For food we did Myanmar Feel - it is a good place to get introduced to Burmese cuisine. We wanted to go to the food stalls on 19th Street but were rained out.
It costs between 6,000 to 8,000 kyat to get from the airport into town. Be careful of them taking your luggage and then asking to be paid for a carry fee. Otherwise no real problems - felt safe.
Mawlyamine and Hpa'an, or the South:
We picked Mawlyamine on friends' recommendations and because we had time to go there. I highly recommend the train, it was the most fun I had during the entire trip. It's also hot, slow, crowded, occasionally stinky, but completely unforgettable.
We stayed at the Cinderella Hotel which is the only up to par place in town. There are lots of guesthouses, and if that floats your boat go ahead. It isn't my speed anymore. Comically Cinderella provides you with a odd and wide range of room freebies, including an angry bird shower cap, ovaltine cookies, packets of powdered cappuccinos and the best shower flip-flops ever (green with caterpillars on them, shaped as a leaf).
We rented bikes from the Breeze Guesthouse and booked a boat service to Hpa'an.
Oh and food. Go to the Beer Garden 2. It's on the Strand, a little walk away from the center of town. It's a Tiger beer garden (as in Singapore beer not animal) where you pick out mystery meats, and sometimes mystery vegetables, from a fridge. They are grilled and served to you with a dipping sauce and rice. It was my second favorite meal in Burma.
We stayed at the Hotel@Tharabar Gate and were all pleased. The rooms were nicely done, the service was friendly, the pool was great. All in all a great place to come back too after rides through the dust and heat of Bagan.
Food options in Bagan are a bit limited but we had two delicious meals at the Starbeam restaurant. I don't do places twice usually, so that speaks for itself. The Rakhine Fish Curry was lovely.
We also ate an an Indian restaurant in Nyauk U (one of the bigger villages in Bagan). Aroma 2 is tasty but they will try to sell you their touring services - the sale is very gentle and can easily be turned down.
Rent bikes, but be sure all the necessary elements are in working order - check breaks and tires. The Horse carts are used by everyone, as a foreigner you are forced to only put 2 people on a cart. It's a little frustrating since you see Burmese families of 8 on them, but that is part of the foreigner tax.
Finally be careful of the kids acting as touts. The number will probably increase and its up to you how you want to handle that. Help them out or discourage the practice - it's ultimately a personal choice.
My favorite temple was by far the Salamani Paya, I loved the wall paintings and the atmosphere.
Pyin oo Lwin and Mandalay:
We stayed at the Kadawgyi Hill Resort and I'm not sure I would recommend it. It was in some ways lovely and in others disappointing. The room was clean, everything worked fine within the limitations of Myanmar (slow Internet, poor English skills, expensive but bare). This was out first encounter with unpleasant staff (again friendly and polite) who refused to help us get cheaper travel back to Mandalay.
There is a very posh hotel next door called the Pyin Oo Lwin Hotel. The rooms are nice (we got a tour) but the place is odd in that Burmese way again. For cheaper places I'm afraid I just don't know.
Bikes are again highly recommended. You can discover Pyin oo Lwin in a day (2 nights) or on a simple day trip up from Mandalay. I am glad we stayed almost 3 days, it was nice to get to know the city a little bit better.
Best meal was a dumpling soup from the Shan Market. If you are coming from out of town and heading towards the roundabout, its the first stall on your left. She has a pot full of it - point and smile. Delicious. Best meal in Burma award.
Mandalay, in my opinion, is skippable. It's too hot, too flat, too decentralized. The palace is a military base and most of the old buildings are gone. The Paya on Mandalay Hill gives a great view of the plain in which the city sites, but that is good enough for me. The U Bein teak bridge was nice because it was cool there, and its nice to see local people out for a stroll, but again missable. The cafes at the entrance to the bridge were frankly dirty. So all in all, if you have other choices of places to see in Myanmar go there instead of Mandalay.
We stayed at the Rupar Mandalar. Out of town a little, very very nice.
General advice on Travel:
Ever wonder why our paper money is so used up and crappy in the states?Its because the Burmese have all of our nice new ones stockpiled. The books and websites do not lie on this topic. Bring all your money with you in 2009 and 2006 (only 100s, no new ones have been issued since then) fresh, crisp and unwrinkled $ bills. The economy is not as dollarized as a year ago, but to get kyats you will need to go and have them exchanged. The exchange desk will inspect every bill, and will give you a corresponding exchange rate depending on the quality of the dollar you hand over. In some cases, but increasing less, they will turn down your beautiful, post-2006, flat, and unfolded $100 bill because of a microscopic discoloration. They really won't take any older money.
This is set to change as the banking system gets linked up to the international system in 2015. Already in a year (we swapped stories with friends who'd been there a year before us), kyats where more widely in use and it was easier to get around with credit cards. Hotels will accept them, no one else yet will.
One final note, don't wait till the day before to source your drug dealer standard USD bills. The banks will not have them at the ready. Go from bank to bank for at least a week before, request specially the bills you need. Increasingly banks are getting used to tourists going to Myanmar and will understand and help.
Get used to Betel nut chewing. The smell, the teeth and the strange red stains on the floor are everywhere. It's not as offensive as it sounds and you quickly get used to it, but the first time you see bright red lips and a mouth with black, red, and missing teeth it's a bit of a shock. It's everywhere. The smell isn't unpleasant but it is noticeable - a kind of slightly sweet and acrid smell, at least to me. The Burmese don't spit about willy nilly, but you do see the effects of it as the ground is stained bright red. It's essentially their chewing tobacco with all the associated health and dentistry problems. It's just also bright red and very wide spread.
Health wise the food is clean, if very greasy and not super vegetarian friendly. Bottled water is easily available and if you are particularly sensitive, western food is within easy reach.
We solved most of our travel problems mostly by standing around and finding that someone would jump in and solve the problem for us. The bike incident in Mawlyamine, transport questions in Pyin Oo Lwin and Mandalay. All were resolved quickly and with very little work on our part. I think that will change as the country gets used to tourists. Maybe it won't too, maybe its part of the national culture, who knows yet. My point here is don't stress, Myanmar has a funny way of stepping in and helping you out unlike any other place I've been too.
My final thought is a simple one - go now, go soon. The country is unlike anywhere I, and many people I have met, have ever been too. It's India and SE Asia, and yet it is neither. It's friendly, open, giving, and genuine in a way no other place I've experienced is. Will that change? Maybe, maybe not. Just don't wait too long to find out.