So what do you think teaching with authority means? It probably means two things. First, much of the rabbinic teaching that we can piece together from near that time seems to have rabbi’s continually quoting other rabbi’s in support. Jesus didn’t do that. He just taught! And, second, his teaching was backed up by power. He was able to cast out demons. Apparently there weren’t a lot of other teachers doing that!
Peter having a mother-in-law does serious damage to the Roman Catholic priesthood. Since they believe Peter was the first pope, him having a wife makes their entire priestly structure unbiblical.
Throughout this reading we see Jesus trying to keep his ministry quiet. One reason is seen here: it affected his ability to minister. The notoriety meant that he had to stay outside in remote places. A second reason has to do with who was doing the witnessing. I imagine that the testimony of a demon might be impressive to some, but it could also be dismissed as a lie. Finally, as it is put so eloquently elsewhere, His time had not yet come. The bigger the impact and notoriety, the more the religious leadership would want to have Him killed. He has more to do until the fulfillment of His mission!
Finally, we again see the example of Jesus taking time to pray. I’ll say it again: if He did that, how much more do we need to?!?!?!?!?