My first question is do you have a bachelor's degree? If you want to fly for one of the major airlines you will need to have one. It doesn't need to be in aviation; can be whatever you want.

So if you don't have a bachelor's, perhaps that should be the first step.

Secondly I am partial to US Aviation Academy of course but one thing I would note is that it is generally cheaper to do your training Part 141, not part 61. 141 however is more structured and takes a bit of a more full time dedication which would probably be easier to do after you finish your service.

After you get your CFI you will probably want to teach for a while to gain hours (and income). Might look at US Aviation for that as well since we just got a deal with American Eagle (see the US Aviation thread in this forum).