I'm not sure what your tax liability is, but I'm going to guess that you're at around $1100/mo take-home. If you have $600 in bills, adding another $240 to that is going to leave you with a whole $260/mo ($65/week) for food, entertainment, shopping, and incidentals - which is probably quite a bit tighter than you'd like.

There's no need to buy a new car; I'd suggest you find something used with some decently low miles on it, have it checked out by a mechanic, and then drive that sucker until it won't go any more. Do your research to find which cars have the best longevity and high-mile maintenance records. Any car you'd consider getting should get a Carfax report run on it before you put any money into it.

You'll get better prices from private parties, but you typically have to pay in cash (or equivalent). Private buyers won't finance you, for obvious reasons. You can get that cash by taking out a loan through your bank, if you don't have it on hand. If you were to buy a $6000 car for $3k down + $3k financed at 4% for 3 years, you'd have a monthly payment of $89, and you could probably pay it off ahead of time, reducing your total cost of the loan. Additionally, you could get a little money out of the car when you're ready to replace it, either via sale or donation tax credit; you won't get that with a lease. You won't get anywhere near what you put into it out of it, but ownership (rather than leasing) can reduce the total lifetime cost of the vehicle.

If you buy from a private party, don't be afraid to get it checked out by your mechanic first. They'll spot stuff that could be very expensive not too far down the road, and let you know what you're buying.