No one on here is going to be able to tell you if your car is ready to go for a trip, because no one here can examine your car.

I would suspect that a car you bought for $500 will not be up for the challenge of a major trip.

But the only way to know for sure is to take your car to a mechanic and let him examine the car. He'll be able to tell you what problems your car has and when you might see them develop.

Even if your car gets the ok for a mechanic, have you considered what you are going to do if you have a mechanical problem during your trip? Do you have money set aside to pay for repairs if you have a problem?

Those are the ways you'll find out if you can safely make this trip with this car.