Some advice on car buying? (Question 1)

I’ve been kind of casually shopping for a car for a year or more. Now I’m getting a little more serious, and I have questions for knowledgeable buyers.

I’m leaning toward the moderately priced compact hybrids. I don’t need a big car, but I do want a dependable four-door with good mileage and one that feels nice and stable on the road.

Question 1. Am I better off financing or writing a check for the full amount? Which will get me a better deal? I’ve set aside enough to make a single payment, but do dealers prefer to see financing? And—in that case, do I lose by paying off the total as the first and only installment?

(Question 2. Question 3.)