Have you been able to get the motor to move with one of the examples from the AccelStepper library?
If not, do that first so you are just focused on a single problem.
It is not a good idea to use the String (capital S) class on an Arduino as it can cause memory corruption in the small memory on an Arduino. Just use cstrings - char arrays terminated with 0.
When using Cstrings you must use strcmp() to compare values rather than ==
And have a look at the examples in Serial Input Basics - simple reliable ways to receive data.
You will make your Arduino program much easier to develop if you just send single character commands. Using the alphanumeric characters you can have 62 separate single character commands. If you need to send a command and a value (such as the number of steps) I recommend using the technique in the 3rd example in Serial Input Basics.