Some fish are super easy to catch (like the Carp, Chub, etc.) but sometimes I get a few that move way too fast. Is it just me being slow, or does it get easier when I level up to a higher fishing level?
The fishes will still be moving as quickly and erratically. However, your fishing bar does increase as your fishing level goes up, so you'll have more of a margin of error in trying to keep the fish within the bar.
It definitely gets easier. CA prided himself on creating in fishing a game mechanic that employed inverse difficulty. Plus, you'll be able to acquire lures and such that make things easier still.