Both of my toilets have recently started making weird noises and the water will start running like it is filling the tank, but the water level doesn't rise. This can happen in the middle of the night when no one has used the toilet for hours or at any time. Any ideas?

It is almost certainly a leak in the flush valve. Most of the time this is a flapper, but it could also be something like this.

Just replace this and you should be fine.
WallyH

Tonight put some food coloring in the toilet tank. Next morning check the bowl. The only way food coloring can get to the bowl is past the flapper and the flush valve. Feel the flapper. If it has wavy edges then replace the flapper. If there are no anhumps, waves or trash on the flat side then the fault must lie in the seat itself. Take your fingernail and run it all around the seat. Any rough edges or nicks? If so you've found your leak. To smooth out the seat, shut the water off at the angle stop and drain the tank. Now take a fine grade sandpaper or sand cloth and smooth out the rough edges. Now turn the water back on and test overnight with the food coloring. Hope this helps and thank you for rating my reply. Tom