A lot of trad shooters do shoot carbon arrows: Gold Tips, Carbon Express, etc. We make up for the light weight of the shafts by adding weights internally.
The advantages of carbon arrows are that they don't bend and are harder to break than other arrows. They are more expensive than other arrows.
The advantages of aluminum arrows are that they are almost perfectly straight when new, and are available in many different sizes. The disadvantages are the noise they make and they get bent and have to be periodically straightened.
The advantages of wood arrows are that they are fun to make and decorate, and are a little quieter in the quiver and when shot than aluminum arrows. The disadvantages are that they are not as straight and there is a greater variation in spine and weight for any given bunch than carbon or aluminum arrows.
You will probably develop a preference for one type or the other as you get more into trad archery. Everybody ends up with a lot of arrows, and ends up using them for something or another. Or else keeps them in a big barrel and uses them for stump shooting or for guests to use.
Far more important than the type of arrow is that you select an arrow with the correct spine and weight for your bow. The cheapest wood arrow, provided it is reasonably straight, will shoot better in your bow than the most expensive carbon arrow, if it is tuned for your bow and the carbon arrow is not.