I'm doing some research on that at the moment. Probably the best short answer is that they tried to do both at the same time, but they made much better progress in North Africa than against the Byzantines.

With that said, the Arabs did make it as far as Constantinople a couple of times in the first century or so of the Caliphate. They were unable to conquer the capital, but serious siege attempts were made. Meanwhile, the success in North Africa was not as sudden and total as most people --including me until about a week ago-- would assume from looking at a map of their expansion. While Arab understanding of desert power was enormously helpful in taking control of northern Africa's Mediterranean littoral, converting the local Berbers was a long, slow process. There were major uprisings that took sustained efforts to put down. It wasn't as total a conquest as one would assume, nor did they make as little progress against the Byzantines as you might think. After all, the Arabs won huge swaths of territory from the Empire in the first century of their expansion. If they put more effort into going east than north in the couple of centuries after that, it can probably be said that they made more progress going east than north, and the Caliphate supported successful campaigns more than unsuccessful ones.

EDIT for Sources: The Arabs in History by Bernard Lewis is excellent. I'm less impressed but also reading God's Crucible: Islam and the Making of Europe, 570-1215 by David Levering Lewis. I find it has a lot of small errors that don't make me question the history but do take away from the overall work. With that said, either book would give you some great insights into your question.