You need a visa. Everyone does, Thai and other ASEAN citizens included. A tourist visa is good for a visit of 28 days. You can get them at any Myanmar embassy, a list can be found here. Three photographs and two visa application forms are necessary. After issue a visa has to be used within three months (at some embassies like Bangkok only one month!).

At the Myanmar embassy in Bangkok a visa takes two days and costs 810 baht. For additional 450 baht you can get it on the same day. It may require a long waiting time, so using the services of a travel agent can save time and hassle.
(I got all my visas from the Myanmar embassy in Berlin, which is fast and reliable. Cost there is 25 Euros plus postage.)

You can stay longer than 28 days in Myanmar. The easiest and most used way is to overstay and to pay 3US$ for each overstayed day at the airport when leaving the country. You can overstay safely for 2 weeks, but as I heard it can be possible to overstay up to 2 months this way. Another way would be to ask at the MTT office near Sule Pagoda about a visa extension.
These regulations might change so enquire before your visa expires.

Update August 2010
From May 1st, 2010 there was Visa on Arrival available. It was 30$US and could be obtained by any nationals on arrival at Yangon Airport. As the elections in November are near this possibility was suspended from September 1st. So back to the embassies for everyone!

The dry and cool season between November and February is the best time.
Hot season is March/April/May, monsoon starts in June/July and lasts until October.
Some statistics from weatherbase.com

Monsoon can be a good time to travel too, as it rarely rains for a longer period of time. Mostly its only short heavy downpours of one or two hours. If this does not bother you and you can cope with the high humidity, the monsoon can be a pleasant time to visit.
(I haven't been to Myanmar at that time, but I experienced Thailand in June/July as very enjoyable. Few tourists, lower prices.)

The usual way to get there is by plane.
You can fly to Yangon from Bangkok, Kuala Lumpur, Singapore, Hanoi, Kolkata, Beijing, Kunming and Taipei. There are also international flights from Kunming to Mandalay.

Bangkok-Yangon-Bangkok is the most popular route, serviced daily by Thai Airways, Myanmar Airways International, Bangkok Airways and Air Asia. Prices vary according to the season, expect around 7000-8000 baht for a return flight (Air Asia about 4-5000, but only 15kg free luggage).

Chiang Mai-Mandalay-Yangon-Chiang Mai was served by Air Mandalay. The route was cancelled in 2008. It is said that the route may be resumed if tourist numbers raise again. Please check the airline's website.
Update Jan 2010: Air Bagan has started a service from Chiang Mai to Yangon and back this month. See the website of Air Bagan

Air China fly Beijing/Kunming to Yangon several times a week. China Eastern Airlines fly from Kunming to Mandalay and back.
Vietnam Airlines fly Hanoi-Yangon four times a week.
Mandarin Airlines connect Yangon and Taipei three times each week.
Indian Airlines connect Kolkata (sometimes also Gaya) to Yangon.
Bangladesh Airlines (Biman) don't seem to fly to Yangon any more.

For up-to-date information check the websites of the airlines in the link section.

Very short summary:
If you are a backpacker and want to travel overland from India/Bangla Desh to Thailand or vice versa by way of Myanmar, you better forget it. It's not possible.
Continue reading if you want details.

Thailand:
Before 2006 this used to be quite easy at MaeSai/Tachileik in the North and Ranong/Kawthoung in the South. I left Myanmar at Tachileik/Mae Sai in 2003 and 2005 without problems.
Following the coup in Thailand in September 2006 regulations became more restricted and it is more complicated now.
To use these border crossings you now need a permit by the state travel agency Myanmar Travels and Tours (MTT) in Yangon. The permit to leave at Tachileik/Mae Sai costs 50$ US plus 15$ for a "guide" at the border. According to reports on the LP travel board, it takes 'a couple of days' to get the permit. You can apply in person at their office in Yangon near Sule Pagoda or use a travel agent. Its easier if you want to leave the country that way, if you want to enter Myanmar you have to get the permit from abroad, which could be possible with the help of a travel agent. Additionally you need a proper 28-day tourist visa issued by a Myanmar embassy.
You can not get visa and permit at the border!

Travelling from Tachileik to central Myanmar is not possible all the way overland. You have to take a domestic flight from Tachileik or Kengtung to Heho or Mandalay as the road from Kengtung to Taunggyi is off-limits for foreigners because of security concerns.
The situation is similar in the south. You can not travel all the way overland to Kawthoung and need to take flights/boats for some parts of the journey.
There are no other crossings from Thailand to Myanmar from where you can travel inside the country. It is not possible at Mae Sot or the Three Pagoda Pass.

The situation at the borders can change quickly, what is written here is based on readings on different forums in early 2010. Check the forums to find out the most recent news. These questions are asked very often, please do a search first before you start a new discussion.

China:
You can enter from China overland at Ruili to Muse/Lashio and continue to Hsipaw. Visa and (expensive) transportation can be arranged in Kunming (Myanmar consulate in Kunming). Alltogether around the same price as a flight from Kunming to Mandalay.
The other way you have to apply for a permit in Yangon (ask at MTT near Sule). For latest information about this border check the LP forum.
Update Spring 2010: According to reports from the LP Thorntree it is currently not possible to cross the border at Ruili

Laos:
So far I have not heard or read of any tourist who has managed to cross the border from Myanmar to Laos or vice versa.
Mae Sai/Tachileik and Chiang Khong/Huay Xai are however only a couple of hours by bus from each other, before 2006 this could be done with a little detour through Thailand.

India and Bangla Desh:
Overland to and from India or Bangla Desh is very difficult and time consuming. The regions near the borders are restricted areas on both sides. You need a lot of special permits and much paperwork is necessary. Only very few foreign tourists travelling with their own vehicles have managed to do this so far. For the ordinary backpacker this is not feasible. If you want details, enquire at the nearest embassies of Myanmar, India and/or Bangla Desh.
A lengthy discussion on Lonely Planet about this can be found here.
There are no boat/ferry connections between India or Bangla Desh and Myanmar, at least no one open for tourists.

Due to international sanctions you can not use credit cards and traveller's cheques in Myanmar!
(Though some of the large international hotels like the Sedona in Yangon can accept them for a hefty fee. Recommended only in an emergency situation.)

The usual way for tourists is to use CASH. You can change Euro, US$, Yen, Baht and other currencies in Yangon, but in my opinion it is best to bring US$ cash. You will need them in any case as you have to pay many services in dollars (see below).

Important: Take clean new unspoilt bills and avoid bills with serial numbers starting with 'CB', as they are often not accepted because people think they may be forged.

Don't worry too much about carrying your money in cash with you. Myanmar is a very safe country for tourists. (This does not mean that you should be careless. Just use common sense and everything should be allright.)

The local currency is called KYAT (pronounced 'chatt'). On some currency exchange websites you will see exchange rates of about 6 kyat for one US$. This is the official rate, which is of no interest for tourists. The inofficial rate in early 2010 is around 1000 kyat for 1 US$. You can check the current exchange rate on this website: irrawaddymedia.com.
You can change US$ for kyat at your guesthouse or hotel, at jewellery shops or other money changers. Do not use the banks.
Do not change at the airport when you arrive! The rate there is very bad. Pay the taxi into town in dollars and change later.

A good place to change money in Yangon is the Bogyoke market (Scott market) in the town center. While walking around there you will be asked if you want to change money. Talk to different money changers to get an idea about the current rate. Best rates are given for new undamaged 100$ bills. You will receive a lot of local currency as 1000 kyat is the largest bill available. One 100$ bill gets you about 100 thousand-kyat-bills. Take your time and count the package, refuse torn and very dirty bills.

FEC:
There used to be a special money for foreigners called FEC (=Foreign Exchange Certificate) which was equal to dollars. Every tourist was required to change 200 US$ into 200 FECs when entering the country. These FECs could be used as lawful money in Myanmar. This regulation was suspended in September 2003. Some older guidebooks and websites still mention it, but it is not in use any more. FECs are however still valid and used by the locals and you may be offered FECs as change, which you should politely refuse to accept.

Paying in dollars and kyat:
Hotels, (most) guesthouses, entrance fees, train tickets, domestic flights and the airport departure tax (10$) need to be paid in US dollars.
Restaurants, local transport like buses/pickups, trishaws and short taxi trips, food and drink and other things you buy in local shops are usually paid in kyat.
Expensive souvenirs, locally booked excursions and guides are sometimes paid in dollars, sometime in kyat. (In my opinion dependent on the current dollar exchange rate).
So you have to figure out a little bit how much money in kyat you will need. Its best not to change too much in the first place. Better to change again when you run out of kyats. You get the best rates in Yangon, a bit less in the other cities.

If you have kyat left at the end of your trip you can try to change it back into dollars at the money changers. Easiest way is to change it with other tourists, or give it as a donation to an orphanage or a monastery. Outside Myanmar you can not change kyat.

No problem bringing your camera, camcorder, cell phone and laptop. They will not be registered or confiscated at immigration as some older websites and books say.

Your cell phone won't work though. There is a mobile phone network in the larger cities available but no international roaming possible at the moment.
You can make international phone calls from many hotels and telephone shops.

There are internet cafes in the tourist areas, but access is often very slow. Many websites like Yahoo or Hotmail are blocked so you may have difficulties checking your mail. Some internet places in Yangon with faster connections offer Skype.
In many internet cafes transferring and burning of digital images to cd's is possible.