Having your tank overcrowded with fishes. This is one of the biggest mistakes newbies make with their new fish pond. It's not about the number of fishes in the tank but how healthy it would be for them.
As others said, overcrowding is a mistake, but also underfeeding or overfeeding. My husband was told by the pet store how much to feed per fish and it wasn't enough so the entire tank died of starvation.
Also, people don't want to take the time to properly clean their tanks and that's a huge mistake.
I myself learned the hard way a lot of don't when it came to my first aquarium and its many things you wouldn't even think of. The first thing I learned was to not overcrowd my aquarium which is something I don't think I would have done but there is always that risk that you may end up doing that if you get over the top with buying fish. Another was to not put my fish directly into the tank for about a day after setting it up, this was something I was unaware of and actually did put my fish in on the same day I set the tank up. Luckily, I don't think it did any real damage but at least I know for future reference.