The best opening move for 6-3 in backgammon is 24/18 13/10. Splits and adds a builder. Watch it on the board, then read the idea behind it.
The idea behind the move
With 6-3 play 24/18 13/10: the back checker gets free, and a checker from the 13-point comes down to the 10-point, from where a 5 reaches the 5-point and a 6 the 4-point.
What comes next
Your next move is to make a point in your home board. If your opponent hits a lone checker, you have other checkers ready to answer, and you lose little.
How to read the move
A move like 24/18 means one checker goes from your 24-point to your 18-point. Points are counted from your side: 24 is the farthest, 1 is the deepest in your home board. “(2)” after a move means you play it with two checkers.