So is there a way to make a program, tells the arm where it needs to go and pick up things, given its initial position? Like let motor 1 rotate a certain degree and then operate the claw and then move motor 2 and 3 a certain degree then open the claw. Can I program the board to make the whole thing automated?
There is way more to accurately controlling a robot arm than you think; note that I am not trying to discourage you in any way - but rather trying to guide you on what you will need to look into in order to have a successful project.
You can do as you suggest; in fact - you need to do that initially - so start there. Get your arm working with the Arduino, first, in the manner you describe.
Note that your method is essentially "open loop" control; while the Vex motors know their position, the Arduino actually knows nothing about the motor positions, beyond simply telling the motors "move here", and then assuming the motors went to the proper position. In order for the Arduino to know the position of the motors, it needs positional feedback from them. This is actually pretty important in a real robotic arm, but for the time being, you don't need to implement it, unless you want to. Just know that in the future, it may be something you want to look into.
Once you have your arm working with the Arduino, unless you are doing simple "pick-and-place" operations with your arm; where all positions are "fixed" in location, and nothing changes (this rarely happens outside of a factory floor, btw), you're going to want more precise control. Ultimately, you're going to want to specify a coordinate position within the fixed work envelope of the arm (that is, all positions the arm can possibly reach), and have the arm's end-effector travel to that position, smoothly and efficiently.
The path the arm takes is known as an "arm solution" (look it up on wikipedia); this path (ideally) takes the shortest and most efficient path, while avoiding any and all obstacles in the way of the arm. Unfortunately, solving for the "arm solution" for a given arm is NOT a trivial problem. It involves knowing all joint angles, lengths of arm segments, as well as the environmental conditions around the arm's work envelope (positions and shapes of objects external to the arm, for instance). It may even need more detailed knowledge about the arm itself (thickness or profile contours of the arm segments, for example). Computer vision processing systems may come into play (involving more than one camera, as well). Then there are the mathematics - know anything about "reverse kinimatics"? If you want to solve for an arm solution - you'll need to (again, wikipedia and google is your friend).
Finally, note that for any given start and end position, the path from one to the other may have multiple solutions (!) - picking which solution is most efficient is not easy; in fact, if you get this far with your arm, you will be quite an expert, and might be able to write your own ticket for future employment at any number of companies.
So - good luck; do your research, and get your arm moving, first. Then take the time to learn, step-by-step, all the other stuff above. Robotic arm control is a broad field of study, with many papers and books published over the last 20-30 years; you would do well to study these.