Not all adventures will be available to you, though, many will come much later. How do you unlock them? Oh, I think you already know. Simply, to get more adventures, you'll have to complete adventures. Yep, you'll have to follow a fairly linear order. Though, from the start of the game, you will still have more than one adventure available to you, so there are options.
The ultimate price
There's one other way to get new characters. You won't like this one. I don't much like this one. Yep, you have to pony up real cash for them. Oh, the humanity! Aye it's a killer, but them's the breaks. You can buy special characters for a small cost. Though, there is an upside. Buying certain premium characters will also result in all ads for your game being removed. A very nice bonus.
Through adventures
You'll quickly discover that most of the characters can be unlocked by playing their respective adventures. Many of them will lives in their own worlds, which are under attack from the Bloons, and you'll need to deal with them in order for the character to join you. Adventures have several stages to them, some with many, many waves, so if you can't do the first you find, try and find and easier one.
Who do you need?
First thing first, you should figure out which character you want to earn. It'll probably be one of your favourites from the show, right? If you don't have any particular favourites, long range characters are usually a pretty safe bet, and will be able to reach Bloons over multiple paths. Take a look at the character menu, and then tap on characters you like, and you'll be told how they can be unlocked.
Order in chaos...