Newby needs some profession advice!

[Lakhi]

You'll get out of Gridania as you continue your Main Story Quests (around level 15). For DRG, Carpenter is a good crafting class to have, since you'll be able to make your spears, but I think Armorer (guild in Limsa Lominsa) would be useful as well for gear. Goldsmith (guild in Ul'dah) makes all the accessories. However, crafting classes are really dependent on each other, and it's most useful to level all of them at once. In reality though, it isn't too important to focus on until you've progressed a bit in the story, since you will need money to craft and that comes with playing the game.

Also, another thing you'll want to start thinking about when you get to Limsa Lominsa is levelling Marauder up to level 15 since you'll need it to unlock Dragoon when your lancer gets to level 30!

[Niwashi]

There's two main reasons why people favor leveling all eight of the crafting classes:

1) Cross-class skills: At levels 15, 37, 50, and 52, each crafting class gets a different skill, but one that can be assigned as a cross-class skill (available to use on other crafting classes once you've learned it on the one it's native to). Nearly all the level 15 ones are really helpful. Culinarian has a really good level 37 one. And the level 50 skills are all important too by the time you reach recipes in that level range. So a good early goal is to get all the DoH classes to level 15 to use that first set of cross-class skills. Then aim for the Steady Hand II skill from level 37 Culinarian. Once you get those, the next useful ones aren't until level 50 which will take quite a bit longer, but an eventual goal would be to get all eight classes up to 50. (I haven't gotten to the level 52 ones myself so can't say from personal experience, but I believe the level 52 skills have their uses occasionally, but none are real game changers.)

2) Materials: Most finished products aren't made entirely from raw materials, but also with intermediate materials that have already been processed. Carpenter turns logs into lumber before turning that lumber into anything from spears to spinning wheels. But the use of those intermediate materials isn't limited to the class that makes them. For example, when a carpenter makes a Bronze Spear it doesn't just use the Maple Lumber that comes from a CRP recipe, but also a Bronze Ingot that comes from an Armorer or Blacksmith recipe, and Fish Oil that comes from an Alchemist recipe. In the earliest levels, you can just buy these materials from an NPC, but as you get into higher levels, you'll need more advanced materials that the NPCs don't sell. At that point, you'll need to get crafted materials, and if you can't craft them yourself, then you're at the mercy of whatever prices other crafters are selling them for on the Market Board. While at times the Market Board is a nice convenience, it can get really expensive to be limited to it as your only option.

For making your own materials, keeping the classes near the same level works best. The items a level X recipe needs can be made by level X (or earlier) in their respective classes. Keeping them near the same level also keeps your gear requirements simpler, since your DoH classes share the same gear if they're at the same level.

For DRG, Carpenter is a good crafting class to have, since you'll be able to make your spears, but I think Armorer (guild in Limsa Lominsa) would be useful as well for gear. Goldsmith (guild in Ul'dah) makes all the accessories.

Plus keep in mind that the crafting classes will need their own gear as well. Weaver makes most of their clothes, along with occasional pieces from Leatherworker. Blacksmith makes most of their tools.