First and foremost-
When buying from a private seller, make sure the vehicle checks out.
Either ask the seller if you can take the car to your mechanic’s own shop or if at all possible, bring your own.

This may be tough to do so if you can - contact a reputable mobile mechanic to come take a look around.

If you purchased from a private seller how did it go? Did you bring someone with you to drive the new (used) car? What about tags? You would need to take the title, proof of ins. etc. to the DMV prior to obtaining tags so how did you get the vehicle to your residence?

Sorry if I sound clueless, but I'm not sure what steps to take and in what order.