If depends on where you want to go with cubing.
If you are looking for another puzzle to solve in World Cube Association monitored competitions, I would say either the megaminx or the Pyraminx.
Solving the megaminx is very similar to solving the 3x3; there is just a lot more to solve. I would only consider it a little harder to solve than the 3x3. Solving the Pyraminx is slightly different than solving NxNxNs (such as the 3x3 and 6x6). It is a very simple puzzle though. I would consider it much easier to solve than the 3x3.
It is different if you are looking for puzzles to solve. By this I mean, you want to figure out how to solve the puzzle yourself. Your aim when solving a puzzle isn't necessarily to solve it quickly.
The axis cube and the 3x3x4 are two good choices. The axis cube is a common puzzle that is in a different shape. Once you find out what you need to do, figuring out how to do it can be tricky. The 3x3x4 looks like a 3x3x3 with an extra layer. It is similar to an NxNxN in how pieces move. However, it solves a little bit differently.
The gear cube would be a good choice too. I'm not sure about the X-Cube though. It may be difficult to solve, but I'm not sure.
Lastly, I would recommend reading this blog post
that someone made.