Jen,
I have been in the car business for over 35 years. 22 as a car/heavy truck/construction equipment mechanic and now for 15 years in management for a large fleet of vehicles.
It sounds like you have the right idea. All used cars should be looked over by a trusted mechanic before you buy. That is the drawback to an auction. They are always "as is - where is" with little or no chance to check anything.
The other thing I have always told people who have asked is: Do not spend your entire budget on a car. If, for example, you have $4,000 to spend, find a car that costs $2,500 to $3,000 maximum. EVERY used vehicle will need some work like tires, radiator, battery, etc. You will now have $1,000 - $1,500 left to spend on repairs and still not be immediately over your budget.
Also, go for an "ugly" car. Desirability will always drive up the price. Don't buy a convertible on a beautiful sunny day - buy it in winter, etc.
Good luck on your search.
Mike