I notice that when I'm quickly alternate picking scales and whatnot on the higher frets, my pick often catches on the strings. I'm sure I'm just using too much pressure and maybe my angle is just a bit off. Are there any exercises or tips that anyone has that could help me improve?
Well having it at an angle helps a bunch but maybe you need a thinner pick. Try to pick with the least amount of resistance possible by having it barely touch the strings and build it up to a comfortable level.
Maybe it could be your pick, like the guy above said. What thickness of picks are you using? I had this problem when I just switched to 1mm picks.

You could also be picking with too much of your pick. Try what the guy above said, but try not to go for the 'as little resistance as little' method. Just try to correct your problem and leave it at that. Some people seem to think using the method could make your playing not have any attack to it when you start to go faster.
I used to have the same problem till i switched to a thicker pick (I found that my cat loves to chew on medium picks and would take bite small notches in the sides that you almost couldn't see, which is why my pick was catching, she can't bite through my thicker picks so I'm safe now...), and also realized that only a very little piece of the pick is neccesary to make the strings sound. I now hold my pick so close to the tip that you almost can't see it. I also found that using this much of your pick makes it much easier to hit squealing Dimebag style pinch harmonics.