Re: Beginning gearstuffs
Welcome.
I would look to the used market with that low of a budget. You'll get twice the bang for your buck there, if not more.
I would suggest looking for used G&L Tribute guitars from when they were made in China (until about 2011 or so, IIRC). There are some that were made with two humbuckers; I would try to find one of those, as they will be quite versatile. IMO, these are the best bang for the buck guitars on the used market. (The fact that they each came with a nice G&L-embroidered gig bag contributes to this. Nowadays the Indonesian ones aren't built as well, and they don't include the gig bag.)
The guitar will run you $250 to $350. I would take the remainder of your budget and just pick up the best $50–$100 amp you can find on your local Craig's List. Then save up for a better amp. I might suggest something like the Fender Bullet Reverb for now. They are cheesy little amps, but they are perfectly serviceable as a stop-gap. Mine served me surprisingly well when I was in the navy; it was tiny, inexpensive, and had a headphone jack. I even played some sideman gigs with it when I was stationed in Charleston, SC.