To enter any country, you will need a passport that doesn't expire for at least another 6 months. If you're going to multiple countries, make sure it's valid for at least another 6 months from when you plan to enter the final country of your trip. If not, you'll need to get a new one from the relevant passport agency in your home country before leaving. It may also be necessary to renew it if it hasn't got many pages left, as each visa you apply for takes a full page, plus you'll be stamped in and out of every country. Getting a new passport can be a time-consuming procedure, so getting it sorted out as far in advance as possible before your trip is recommended. You need to do this before applying for any visas, if necessary.
You may need a tourist visa depending on the countries you're going to, your nationality and your intended length of stay. A visa is a sticker or stamp in your passport, applied for at the embassy or consulate of the country you want to go to, that shows you've been authorized to enter that country. Visas are typically applied before either in your home country (not always possible as not every SE Asia country has embassies everywhere), or in a neighbouring country to the one you want to visit. Visas rules can and do change often, and the ease of getting a visa varies from embassy to embassy. This is even true for different embassies of the same country, as how stringently the officials apply the rules differs everywhere. Applying at a consulate is generally easier than at an embassy, if possible. A visa always comes with an expiry date - note this is not how long you're permitted to stay in the country, but rather the when the visa must be used by (i.e. you must enter the country on or by that date).
Visas are usually single-entry; that is, they're used up when
entering the country. If your travel plans are more complex and involve going to
the same country twice (or for a period longer than a single visa allows), it
may be better to to apply for a multiple-entry visa instead.
Getting a visa-on-arrival is possible sometimes too, this involves filling out a form and paying the visa fee only entering the country, so it's more convenient than having to apply in advance. Visa-free entry is better still, as you don't need a visa at all to enter if this is the case. Both visa-on-arrival and visa-free entry are generally for shorter stays, and applying for a visa in advance will let you stay for longer.
Visas should generally be applied for before booking your flight, so as not to risk paying for a flight you'd then be unable to use should your visa application be declined. Applying for a tourist visa is mostly routine for anyone from a Western country though, and you won't be declined barring exceptional circumstances.
A visa itself does not directly give you the right to enter or stay in the country, as that is solely determined by the entry stamp stamped in your passport by the immigration authorities when you enter that country. The entry stamp marks the date you arrive, and will tell you the date you have to leave by. How long you're permitted to stay varies by country, nationality and what type of visa you have (if any).
When checking-in to fly out of your country's airport, the airline staff will check your passport for a valid visa and/or return ticket if they're required for the entry to the country you're going to before letting you board. It's not unknown for items to go "missing" from checked baggage along the way, so make sure all your bags are locked as much as is possible. Any items of value should be only in your hand luggage. Once you're in the air and approaching your destination, airline staff will distribute the Immigration form for the country you're going to. You'll need to fill this in with your details regardless of your visa status, and present it to the Immigration officer along with your passport when you land.
When your flight lands at the airport, the typical procedure is that you are first directed to pass Immigration. The one exception to this is if you need a visa-on-arrival (not visa-free entry), in which case you'll need to follow the signs for that first and get the visa before heading on. At Immigration, the officer takes your passport, your filled-in immigration form that was given to you on the plane, examines your visa if any and will stamp your passport with today's date and the date you have to leave by. Immigration officers are only human and will occasionally make mistakes, so it's worth being aware of how long you should be stamped in for and checking it's correct before moving on.
Next step is Customs, where you will have to declare or not anything dubious you have in your luggage. Customs requirements are mostly common sense, and similar to requirements that airlines also enforce - no guns, weapons, explosives, drugs, hazardous chemicals, live animals, plants, etc. You shouldn't also bring in any single item in large quantities, as it may look like you're trying to evade import duties and sell it. You should bring the prescription for any prescription drugs along with you, and if possible try and keep any medicines in their original packaging. South East Asian countries have some of the world's harshest drug laws, so be wary of taking anything that could be misinterpreted as illegal drugs, such as unbranded or anonymous-looking pills or powders.