This is the same for the Weaponsmith profession as it is for the other professions. While running around in the world, Gather wood from any tree you see, and mine ore from every rock you see.
Obviously, not every rock and tree is available to gather resources from; interact-able trees, ore veins and vegetables/spices will be marked on your minimap with their respective symbols. A small tree for wood gathering spots, a rock for ore gathering spots, and a grass icon for spices and vegetables. The last one (vegetables/spices) is not relevant for the Weaponsmith profession.
If you find you are running out of inventory space (don't worry, crafting materials stack), simply open your inventory, click on the cog icon in the top right part of the inventory window and press "drop all collectibles". This will send all crafting materials and other collectibles in your inventory directly to your bank, in their own separate tab, dedicated to keeping track of the materials you own, so you don't need to worry about your bank space clogging up with crafting materials.
BEWARE before you press this button though! Pressing this will also send any minipets you currently have in your inventory to the bank. To prevent this, simply send the crafting materials to your bank manually by right clicking on them and clicking "drop collectible".
As for crafting materials other than ores and woods, you will need to kill monsters, lots of them.
The most efficient way of gathering materials such as Vial of Weak Blood and Pile of Glittering Dust is partaking in events; events tend to spawn extra monsters, and you will get the chance to gather a lot of materials by simply getting in one or two hits on each monster in the area, therefore taking part in their killing and earning a chance of them dropping loot for you.
SACKS tend to also be a very efficient way of gathering the materials, since they drop easily and almost always contain craftable materials. Sacks are most commonly dropped by enemies that have the ability to speak, or display some measure of civilization. Centaurs and Ettins are examples of this.