Staying Riding After the Boardstart
It's very important to seperate boardstarting from riding. They require different skills. The riding process only starts after a successful boardstart. If you try to do both at the same time, you probably won't do either.
Once you are on top of the board, keep the kite moving in long powerstrokes. Make sure the nose of the board it higher than the tail of the board, so if you sink, the board won't submarine into the water.
If the board sinks, that's ok. Just make sure your next powerstrokes are powerful enough to get you out of the water. Keep the board pointed at the kite.
If you repeatedly sinking, but you are still on top of the board, try edging the board a little bit away from the kite on the downstrokes. This should help you pick up speed.