What are the differences between A and B? I think both mean almost the same thing.
I often went to various places once I had bought a new car.
Similar questions have been asked before, although they may not answer your question completely. Anyway, please read them first:
Now you understand the cases where 出かける is absolutely wrong. If you won't go back, or if you go to somewhere against your will, 出かける cannot be used.
But in your example, the speaker went out somewhere on his own will, and he was going to return home eventually. So are Sentence A and B completely interchangeable? Well, I feel there is a small difference, but it's very small.
出かける is basically for relatively short and casual outings. If I hear よくいろいろな所へ出かけた, I would imagine the speaker used his car a lot, but mostly for going to places within a few hours distance from his home. (Actually, some people may use 出かける when they go to places hundreds of miles away from their home. It probably depends on how casual the speaker thinks the outing is.)
On the other hand, 行く has broader sense. If I hear よくいろいろな所へ行った, I would imagine the speaker traveled to various places, including those which are far away from his home.