I'm sorry to the OP; I totally forgot to address your post. I would say yes, you can get a good vehicle for around $3,000. I've seen them as low as $1,500-$2,000, but you really need to look them over well and ask a LOT of questions. If the seller seems shady, doesn't know a lot, can't tell you when the oil was last changed, etc. then walk away; there are other vehicles that you would be much better off with. The best vehicles I owned were bought from the first three owners, and had maintenance records; nothing fancy, but at least it showed a pattern of routine maintenance, which is one of the things to look for.
I'll give you some tips; but this is by no means comprehensive; that would take all day, and more experience than I have by far.
Check the tires all around; not just for wear and weather cracking, but is the wear even? If not, it could be a sign of an alignment issue, or something worse. If it has an automatic transmission, put it into drive with your foot on the brake, then put it into reverse, then repeat; does it make a loud cluck or is it a smooth shift? If it's a manual, where is the clutch petal at when it releases? When you push the clutch in, does it make any odd noises? Does it shift smoothly through all the gears?
Check for oil leaks, not just the engine, the transmission, differential, etc. Check the tailpipe for any signs of it burning oil. Make certain it doesn't smoke, as that's a sign to run away. Check the oil and coolant levels. Are the hoses pliable? Do any of the belts have cracks?
Test all of the features to make sure they work, as any repairs will add up, especially if you can't do them. If the paint is different colors or has primer, was it in an accident? Check all the gaps around all the body panels to look for signs of prior collisions; if things look uneven, it probably was in one.
Does it idle smoothly? Check the date on the battery and look at the terminals for corrosion. Does it drive straight doing down the road? Does the steering feel smooth? Are there any wobbles with the steering? How about when you are braking? Check the heater and AC. Don't run anything loud while driving, and keep the windows rolled up, as you need to listen for anything that sounds odd.
I'm sure I have missed a TON of other things to check, but that should get you started, and I would also suggest taking any vehicle to a good mechanic that you find (not one the person selling the car recommends), and have them check it out.
I hope this helps and I wish you luck in finding a vehicle that is right for you and your travels!