The Takeoff
The takeoff will have two key differences from a normal takeoff. One, you will need to turn the ailerons (yoke or stick) into the wind before you begin the roll to help the upwind wing from being picked up by the wind. This “crosswind correction” will slowly be taken out during the takeoff roll as the ailerons come almost all the way back to center just before the wheels leave the ground. The reason for slowly taking out the correction is that the ailerons become more effective as the airspeed increases during the takeoff roll (meaning you’ll need to slowly work your way back to neutral ailerons, otherwise you may actually develop enough force to push the upwind wing dangerously close to the ground).
Second, you may use slightly less back pressure during the takeoff roll, and rotate perhaps a few knots faster than usual (3-5 knots in most GA aircraft) to ensure you do not become airborne at a speed where your flight controls will not be able to overcome the forces of the crosswind at the moment you transition from being a propeller driven car to a flying airplane.
Everything else will be very similar to a normal takeoff, rudder to maintain centerline, checking engine gauges immediately upon applying full power, etc.
It is also important to note that you will be taking off in a “cross-controlled” state. I.e. strong right crosswind would require you to rotate and begin flying with left rudder and right aileron inputs. Right after lift off, you will want to transition back to your normal take off procedures, allowing the airplane to weather vane (crab) into the wind to track the extended centerline of the runway, and using appropriate amounts of right rudder to counteract left turning tendencies while maintaining wings level (as close to 0 degrees of bank as possible). Remember, wings level gives you maximum vertical lift with no horizontal lift.