how does crafting work on this game and which crafting is good to do
You get the recipe ( either by discovering it or by purchasing it from the tp or an npc) and craft the components to craft the item assuming you have all the necessary materials. An extremely simplified explanation. As for which is good, depends on what you mean by it. Good as in most profitable, or good as in gets you the gear for your class.
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Below Ascended quality gear there is no real point in doing any crafting as everything can be bought faster and cheaper from the Trading Post. You also get plenty of lower quality armor and weapon as drops while leveling up to 80. Chef used to be special case, able to create good quality food, but there are now even better sources so there is no reason to level up Chef anymore.
Now Ascended and legendary gear is non-tradable so every account has to craft these for themselves. If you have all the classes and want each to wear best quality gear, then you must level up every crafting profession, that means 3 different types of weapon and 3 different types of armor crafters. Leveling up crafting really means buying every material from Trading Post and then clicking the button and waiting the character to finish all the tasks. All you need, is lots and lots of gold (which can be easily converted from gems which can be bought using real life money). Or you can level up your crafting in old way, gathering materials yourself, making few items and breaking them down again to restock some materials, this can take very long time to max up your crafting.