The difference between E-1 and E-3 ~$270/month. At this point, I'm not sure if you've actually done the math behind this, or if there's something you haven't posted. Here's what I get: according to the info I've found, you don't get BAH or BAS during BCT and AIT, so (assuming you're an E-3) your total pay is going to be a bit more more than $1800/month. Assuming an annual 1% raise (based on past trends), let's round it up to $1850/month. 55 weeks is 13.75 months, so total pay for basic and AIT is $25,437. A quick tax calculation shows ~$2400 in federal taxes--hopefully you live in a state with no income tax, but if not, then you'll have to factor that in.

So, assuming you want $20k at the end of this, that leaves ~$3000 for a year's worth of expenses. My brother went through OSUT earlier this year, and I believe there were a few deductions for uniforms and whatnot, so you'll have to subtract that out. Then you mentioned car insurance. What about phone service? Saving for something other than travel? How about all the small things you'll need--toiletries, clothing? All that said, I don't think it's unreasonable to put away a good amount for travel (maybe $5k?).

As for where to go, 84fiero pretty well covered it based on the info you've posted. To get any more detailed info, you need to figure out what's important to you--do you want to see ancient history? More recent history? Language? Art? Otherwise, you're more or less asking for a mind reader. But again, you've got plenty of time to figure out that sort of thing, so good luck.