What is the best safe and cheap country to visit in Europe in summer?

Pay attention to Germany, Poland, Portugal and especially Bulgaria is a great option for summer vacation. All these options are affordable so if you haven't been there - I highly recommend. Also, check out this article https://travelsites.com/blog/how-to-get-affordable-last-minute-travel/

This is the topic I've been looking for quite a while for I am always interested in the budget travel. But please tell me whether I can take a car from the U.S. and go to the destinations you mentioned?

It dependson where you want to travel. I don’t think you can travel by your car right to
Europe. But what I would suggest, for example, is that you drive your car until
some distant airport, it can be JFK in NYC or something else. Then you take a
plane and travel further while your car will be taken care of by https://www.sgtautotransport.com/services that will safely ship it back home.