Relic weapons: There's two kinds: Zodiac weapons are the level 50 ones, and Anima weapons are the level 60 ones. Rinari did a decent job explaining them, but to go a little more in depth, they require a questline to create, and are notorious for the amount of grinding required. While older steps have been reduced in difficulty since being introduced, the fact remains that they take far more work to create than a weapon of similar power. The main advantages of a relic weapon are prestige (It's a sign of dedication), and that it's the only way to get an "endgame" weapon without stepping into raiding or extreme mode primals, albeit at the cost of taking far more time. You also get to choose the stats on the relic, which is good for those who aim for specific substats. So, all things considered, I wouldn't go for either of the current relics currently unless you want to just do them for glamour/prestige, but I would recommend going after whatever relic they add for 4.0.

Poetics tomestones: As Rinari mentioned, these are used to obtain ilvl 120 gear. However, they can ALSO be used to obtain upgrade materials to make that gear ilvl 130 (and in fact a set of ilvl 130 gear will make you start off Heavensward in prime condition, you won't need any gear upgrades until well into it). Furthermore, there are an assortment of other materials you can purchase with them, such as crafting materials or items used in relic quests. Lore and Scripture tomestones are the current two endgame tomestones, allowing you to purchase equipment for level 60 characters, with Scriptures being the highest level and lore being second best. Whenever the item level cap raises with a new cap, they introduce a new tomestone for the new item level, the previous best tomestone gets downgraded to second best, and the previous second-best tomestone gets removed from the game and the items previously purchased with it become acquirable through other means such as grand company seals or centurio seals. In any case, the highest level tomestone is only used to acquire gear, and has a weekly limit of 450 able to be earned, which limits how fast you can acquire the best gear. The second-best tomestone has no such cap, allowing you to get the second-best tomestone set as fast as you please. The second-best tomestone, like poetics, are also used to acquire a number of miscellaneous items.

Grand Company missions are fairly simple. Basically, you want to train your squad three times a day and send them out for missions once a day to level them up. Every so often there will be a difficult mission you need to do to rank up. The main advantages for these missions are the first lieutenant upgrade which unlocks more items you can buy for GC seals and raises your cap, the scrolls from the highest tier of missions that essentially function like personal FC buffs, and the various goods you can obtain from Squadron Chemistry such as materia or crafter/gatherer scrips.

Gathering classes can indeed be boiled down as just there for the gil you can get, though for plenty of people like me, there's satisfaction in having gear you created completely from scratch. I do wish there were untradeable items one could gather, such that you can't just replicate having a gatherer by having a lot of gil... but I know whenever I suggest untradeable items for crafters/gatherers, I get a lot of pushback from people who feel the main purpose of gathering/making items is to sell them. Kind of a catch-22 really...