From the beginning you have access to all 3 titan classes through the given default loadouts (Atlas, Ogre, and Stryder). Once you unlock the ability you customize your own titan loadouts you only have access to the Atlas chassis.
In order for you to unlock the Stryder and Ogre chassis you must play the Titanfall Campaign 2 times; once as the IMC, and then once as the Militia forces. Titanfall’s campaign isn’t a traditional single player or co-op story mode as you would expect from a FPS. It’s nearly identical to the regular multiplayer experience, it’s 6v6, it features either attrition or hard point, and plays on 9 of the multiplayer maps each representing a mission. The only difference is that the story is delivered during the pre-game lobby as audio, there’s a longer cinematic beginning to the matches, and video and audio cues appear in your HUD throughout each match. The story and cinematic prologues do differ for each side so play it twice to get both sides of the story.
I highly recommend that everyone plays the campaign first, even though you might be tempted to jump straight into regular multiplayer. You still earn experience but you’ll get access to the other titan chassis along the way.
After you beat the campaign once, you unlock the Stryder titan the nimble fast paced titan. And then after you beat the campaign a second time you unlock the Ogre titan for customization.