For a good catapult, you want to have the end of the catapult arm move as fast as possible.

The longer you make the catapult arm, the faster the end will move. But, if you make the catapult arm too long or too heavy, it will be harder and harder to accelerate the arm in the first place. To build a good catapult, you want to try different arm lengths and find the one that throws your object the farthest.

Another very important thing to think about is at what angle you stop the catapult arm. The best point very much depends on what the catapult is launching - how heavy it is, and how much air resistance it feels - so you have to try different points and see what works best!

There is no "one best way" to build a catapult. One has to consider the goals and the available resources (e.g., materials, time, tools). Here is are a few pages on catapults and their physics that might help.

That depends on the size of the catapult that you want to build. For small catapults, the onager design is generally the best. For larger catapults, the trebuchet becomes a better design.