Hi, Welcome to the forum.
Look into the Servo.h library in the Learning Section, go from there.
Do not power your servo(s) from the Arduino - provide a separate power source.
Do connect all Grounds together, need that common reference point for signal levels.
Same with DC motors - do not connect directly to Arduino pins. Use a NPN transistor or N-channel MOSFET to sink current thru the motor; use an H-bridge if you want two direction control.