I bought my first car from a private seller as well since then I do trust dealerships only. As a rule they don't sell bad history cars and give some guarantees. If you've already made up your mind at least take it to independent mechanic. Or lookup car by vin: https://trustvin.com/auto-history-by-checking-the-vin-number there are lots of other alternatives like autodna.com, clearvin.com, nadaguides.com Choose what you like, compare prices.
Still I strongly recommend not to buy from a private seller!!!
ask for timing belt service records if the car has miles on it... if they can't produce either walk away or know that's going to be high priority and possible point of $$& negotiation