A nice place? depends what you think a nice place is. You can browse
here to give yourself an idea.
$1 USD is about 37 Thailand Baht
Expect to pay a lot for decent internet speed, you'll rarely get download speeds above 100 kb/s regardless of the connection you have - all connections are shared and it's hard to get decent speed.
Also expect to pay 2 or 3 months rent security deposit plus the first month up front (sometimes for some properties, you might be asked to pay 3 months up front).
If you can eat Thai food and not so concerned about eating off street shops and small restaurants, you can have a good meal and a drink for about $1.50 $2.00
If you stay in a hotel in Bangkok, I can vouch for
this one. Or if you're looking for cheap place to stay with free wireless internet you can try
here - it's about $18 per night.
You'd probably want to get around with a cab in Bangkok, they are everywhere and they are pretty cheap (expect to pay from $1 to $4 depending where you're going in town). The skytrain is pretty cool too, but since I have my car here I never really took the skytrain, just one time to test it out so I don't know how much it cost... pretty cheap tho.
I wouldn't suggest to get a car here unless you know where you're going, parking spots are inexistant and the traffic jams are permanent.