For trying it out, an English saddle is perfectly fine.
But if you do want to get more serious about it in the future, then I would invest in a barrel racing saddle (not just "any" western saddle). Just like a dressage English saddle is different than a jumping English saddle, a barrel racing saddle is designed specifically for the sport of barrel racing. The seat is going to have a higher cantle to "hold you in". The horn will be small and tall so that you can grab onto it during a turn to keep yourself out of your horse's way. The stirrups are also going to be slightly forward to keep proper position during a turn. There's going to be less leather on the skirts for less weight. Etc etc.
If you just want to have fun with barrel racing and experiment with it, you can barrel race in any tack you want. There are no tack rules. People might give you a funny look but who cares if you are having fun?
∞•*˚ Βгįťţαňγ ˚*•∞
It is not enough to know how to ride; one must know how to fall.