Introduction/Moving to Bangkok

I don't have all that much advise to pass on but it is sad this discussion didn't get further.

As to where to stay the obvious answer is anywhere along a BTS line, but in that case anywhere along the airport link train line might work too, since it ends at the Phaya Thai BTS station (more or less, they link there).

It might be the case that you are more concerned about a neighborhood seeming a certain way than the kinds of budget issues people tend to have.  A neighborhood area like Ari (a few stops along that BTS line) is popular with expats; maybe that's an important concern.  Chinese and Japanese expat friends both lived in that area, but I don't think their concerns were about minimizing apartment costs.  All along Sukhumvit is where expats (foreigners) tend to live, all the way down to On Nut or so, and I guess there's no reason you couldn't live further out.

As to work if you have a college degree and can teach English that will work, otherwise things get difficult.  It seems really problematic to move with a baby who is half a year old and then go to work.  I'd trust my own mother with a child of ours who was that young but not many other people at all, certainly not someone who some random foreigner recommended as likely to be reliable.

Networking with people who are either living here posted in a similar fashion or have been would seem the best approach.  They could compare their own experience with other experiences you are probably already familiar with and pass on great advice.  It probably wouldn't hurt to live near where some other people in similar circumstances to live to jump start social networking a little.  I'm just not networked myself, which also works, but that approach isn't for everyone.

I've got kids who are 5 and 10 now so I can probably address some other specific concerns that might come up.  It seems obvious enough but if you are using formula I'd pack a suitcase full of that, just in case.  There probably are US military related shopping options here that would circumvent normal concerns about brands and options differing, so it probably wouldn't be necessary at all, but you changing deodorant  or shampoo to a really unfamiliar style is inconsequential compared to what's serving as food for your baby.

Related to language issues, the first time someone doesn't have any idea what you are saying it will freak you out but beyond that it's nothing to worry about.  Lots of people speak English in Bangkok, even down to the level of taxi drivers (but only a little), and signage in English is a given here (funny how that changes things, in countries like Russia and China), so it won't be a problem.