Also, how does group play actually work? For example, on these fetch quests and stuff, is that done as a group? Or are those individual, and you just take care of the large dungeons as a group? And when can I expect to see those? With the archer I made it to the forest.
Pretty much any quest can be done solo or in a group. Even dungeon quests can be done solo.
If a quest is a "solo quest" (indicated by
not having a little blue person-shaped icon with the number 5 beside it), you can pretty easily solo it with any class as long as the target mobs are not much more than maybe 2 levels higher than you (considering the quest distribution in the game, it is nearly impossible to encounter quests too high for your level).
If you group up with anyone else on the same quest, you will share progress on it. If you need to kill 10 of some monster, you killing one of those monsters will count for both of you as long as you're not too far apart.
If it is a "kill monster for X quest item drops (that don't actually take up inventory space)" then the drop rate is the same for everyone in the group, so if you kill 4 mobs and got 2 of the quest items, everyone in the party on that quest will get 2 of the quest items. There may be exceptions to this, but I've never encountered any.
Quests to gather nodes (the flowers, for example) are slightly in that some things you gather yield a random amount of stuff. For example, the Cloud Mushroom quest near Lumbertown. Those can yield 1-3 mushrooms per gather, so if you gather those as a group, then someone may finish before everyone else.
Escort quests also can be done as a group. When someone in the party starts the quest, it will start for everyone else nearby in the party who is on that quest, too.
Group quests (BAM quests and dungeon quests) work the same way as normal quests outlined above, but they are normally designed with more powerful enemies for a group to tackle. At the early levels, BAMs can be soloed easily by many classes if you don't suck. The higher level you get, though, the nastier the BAMs get, and even the ones you can still solo generally have so much HP that soloing them is a PITA. Hunting BAMs as a group is best, but you don't always need a full group for BAMs. Last night on my 47 lancer I was duoing level 46 BAMs with an archer and making decent time at it. No healer. No deaths. Just have to play smart when you don't have that safety net (healer). Likewise if you're in a group with no real tank.
First dungeon and BAMs are around level 20. If you follow the quest hubs in order, the quests will lead you right to the dungeon area and give you a stack of quests to complete in the area.