Yes, it's a great idea to travel by car. Personally, I can't imagine my trip without a car. That's why I found a good service that provides car rentals all over the world. Or if don't want to drive by yourself, you can order transfer. I always travel with Gettransfer.
Oh, if you are going to Paris, I have an advice that you definitely should follow. The best thing I've experienced in Paris was a nuru massage and you must also try it, believe me, you won't be disappointed. And I agree with your statement about traveling on a budget. Especially talking about Paris, this city is quite expansive and you need to have a lot of money to eat in good restaurants and visit all the museums that are interesting for you.