Your problem is that you're trying to fight as a Carpenter. All crafting and gathering classes have almost no combat ability, although they can use Wrist Flick to cause their next Throw Stone to stun for escaping.

As a Carpenter, you can do local leves and non-combat sidequests (there are quite a few), or even go explore the world on foot if you're careful. Even so, you will want to pick a fighting class, either a Discipline of War or of Magic, and pick up its starting weapon. As vax says, NPCs sell these in every city, and they cost around 1k gil if I recall correctly.

You should also get familiar with Yellow Gremlin, a very detailed third-party database. The link below will list all the weathered weapons and the NPCs that sell them. Click on the NPC name to see their location and find the one in your starting city.

http://ffxiv.yg.com/items?f=s=weathered;c=1