Fletch to your heart's content.
The woodworker is the premier artisan class for producing arrows, ranged weapons, shields and wooden weapons. Arguably, the best class to pair this with would be the Ranger, as they need a constant supply of high quality arrows to use most of their most powerful abilities. Even if you aren't a ranger, those that are provide a steady market for your goods while you sell the less replaceable items more occasionally. Woodworkers chiefly need a lot of wood to put together their goods - obviously. This is a resource that goes in tiers, with the higher level wood types only available in high level zones. While it's possible for someone with a low adventuring level to go out there to gather wood, it's far easier if you either buy it from other players or gather it yourself.
Keep in mind that woodworkers can only create wooden shields - metal ones are crafted by armorsmiths. They can also craft musical instruments, which are high in demand for bards of all sorts looking to improve their abilities. They can also create wands and staves, items that mages and some priests are looking for. The totems that they create fit in the charm slot, and can give a number of beneficial utilities. Many of them also shapeshift the user into a fun looking monster or creature, adding to the flavor of the items. For example, Spirit Totem of the Goblin shapeshifts the user into a goblin and increases out of combat power regeneration by 96 - a great ability for reducing downtime in between kills. Spirit Totem of the Cockatrice shapeshifts the user into a cockatrice and increases their health regeneration by 228. Both are only usable by characters level 70 and above, but there are many lower level versions that woodworkers can make as well.
The mainstay of a woodworker's business is likely to be arrows, followed closely by totems, shields and weapons of various types. You should try to broaden your recipes as soon as you can so that you can serve more markets. It helps to advertise your services so that you can create things on commission rather than blindly producing items - unless you know that there will be a demand for it. You don't want to end up spending platinum on expensive materials for a weapon only to discover that nobody really wants it. People are always looking for good crafters - build relationships with classes that use woodworking items to increase your business.