Get your finger in every pie you can possibly reach. It will do nothing but good for both your riding and your horse.
I regularly compete in:
And I also do, just for fun and general riding:
PC rally days
Seriously, there is really no drawback, it will make your riding life much more fun, and will make for a more well-rounded horse and rider.
I agree Wild Spot.
I think having a solid basis in one discipline and trying out others is an interesting way to go.
I actually started out as a jumper. Switched entirely to dressage but then also dabbled in western games and pleasure, driving and hunter.
If you get the right horse you can do this easily, with a more set horse, no not so easily.
Personally to me it really matters how versitile your horse is that will allow you to explore other disciplines.