It may be a bit pricey to get to Thailand by air, but once you're there you can take advantage of bargain-basement flights. Just bear in mind that flights in and out of Thailand are often overbooked so confirm, confirm and reconfirm. Buses are a sterling way to get around - they're fast (often terrifyingly!) air-conditioned and comfy. There are even women-only buses. However, there have been bad reports of the service on buses booked from agencies on Thanon Khao San. If you want to get to Malaysia, there are train services.
The bad news is that it can be quite expensive flying to Bangkok, depending on your point of departure; the good news is that once you're there you can shop around for an inexpensive return ticket. A host of international carriers land at Don Muang, Bangkok's major airport terminal. Flights in and out of Thailand are often overbooked so it's imperative that you reconfirm ongoing flights as soon as you arrive. The departure tax on international flights is waived if you're in the country for less than 12 hours.
Overland travel from Malaysia is popular and there are four border crossings between Thailand and Malaysia, two on the west coast, one in the centre and one on the east coast. It's not possible to buy through-fare tickets for rail journeys between Bangkok and Kuala Lumpur or Singapore, unless you ride the luxurious Eastern & Oriental Express, but the trip can be made on express trains via the Thai-Malaysia border at Pedang Besar. The journey usually requires an overnight stop in Butterworth (Malaysia) in order to comfortably make train connections.
There are plenty of crossing points between Thailand and Myanmar, Laos or Cambodia, but very few border crossings are made - officially, at least.
It's legal for non-Thai foreigners to cross the Mekong River by ferry between Thailand and Laos at the following points: Nakhon Phanom (opposite Tha Khaek), Chiang Khong (opposite Huay Xai) and Mukdahan (opposite Savannakhet).