In the former days, Myanmar is accessible by air and by waterway to Yangon, ex capital. Then, Mandalay, second capital is also accessible by some international flight via Thailand, too.
As Myanmar has 5 neighboring countries: India, Bangladesh, China, Laos, Thailand, there are the bordering towns, among them some are allowed to enter to the foreigners.
Before 2013, the border crossings with the neighbors were opened and closed from time to time. But, the situation has changed since the 28 Aug 2013 and foreigners can enter Myanmar by land from at least 4 Thai/Myanmar crossings, 1 China/Myanmar crossing and 1 India/Myanmar crossing.
According to the Immigration Board of Myanmar, it is also possible to go out of the country on the land border.
From Thailand :
You can cross this border to travel into Myanmar, but you will need to take domestic flight to continue the other parts of the country.
Mae Sot/ Myawaddy
You can cross this border to travel into Myanmar. There are buses to continue to Yangon passing through Hpa An, Bago.
For the direct bus, driving hour will need around 12 hours.
Htee Khee/ Phu Nam Ron
This is the most direct border from Bangkok into Myanmar. The crossing is located 66km west of Kanchanaburi along the 3512. The Myanmar side is a wide 4 lane gravel road through the natural jungle, it only takes 4.5 hours to arrive in Dawei or Maungmagan Beach.
Sangkhlaburi/ Three Pagodas Pass (Payathonzu)
This crossing is reopened since 2010. The foreigners can access inside the country and legal trade with Thailand from this pass is allowed.
You can cross this border and can continue by bus, boat or plane into the rest of the country. But, the road is quite bad because of the heavy rain and flight is the best way to reach the other part.
From China :
Foreigners can enter Myanmar at Lashio via Ruili (in Yunnan), you will need a permit (as well as a visa) and a guide on both side. You will recommend to join an organized tour. It's possible to fly from Mandalay to Kunming, and there's a Chinese consulate and can issue visas in Mandalay.
From India :
Tamu/ Moreh crossing is opened since August 2014 and it is possible to cross this boarder with a certain permit from Myanmar side to enter and exit. The necessary permit is available in Yangon, through some agencies.