I find a little confusing that the game has so many quest types, and I can't get what are each type of quest for.
There are Caravan quests, which as it seems, are the quests that progress on the game story, and this one is pretty straight forward.
There are also those quests at the Gathering Hall:
Low Rank Quests
High Rank Quests
Arena Quests
Challenge Quests
Guild Quests
Event Quests
Expeditions
And when I talk to the Palico in my house, there is also an "Episode Selection".
Could someone explain to me what are each of these for, and why would I want to do a quest of that type?
Best Answer
It can be quite confusing for a beginner ;)
Solo
Caravan Quests make the main story ; you can only enter them alone, and they tend to be easier than their Gathering Hall counterparts. (1* - 6*) are Low Rank, (7* - 10*) are High Rank, which means you obtain higher tier materials allowing you to craft better equipment. These quests are accepted from the Guildmarm, and more are unlocked as you progress through the story and complete Gathering Hall quests (see below).
Expeditions are set in a special zone, the Everwood, and allow you to easily gather special items and relics. They also give you Guild Quests and Caravan Points, depending on how many monsters you kill and how many monster drops you get. Expeditions are started through the world map, and come in both Low and High Rank varieties.
Online
The Gathering Hall offers you many quests. You can play with friends (or just anybody, in MH4U), which is nice! Or, if you have no friends nor internet, you can do them alone (my life is so sad, sometimes). These are typically harder than their Caravan Quest equivalents as the monsters within have higher HP (source: reddit, personal experience), as they are designed to be tackled with a team. All the following quests are started in the Hall.
As in the Caravan, there are Low Rank (1* - 3*) and High Rank (4* - 7*) ; there are also G-Rank quests, which are much harder, and give you even better items.
Guild Quests are gained from successfull Expeditions, and via StreetPass ; they have to be registered before you can undertake them. Each time you complete a Guild Quest, it "levels up", which means the monsters will be stronger. Here is the equivalence between Guild Quest level and Rank :
In the Arena, you have to defeat monsters the fastest you can, with the provided equipment ; then, you can compete with your friends for the best time. And you gain coins, used to craft specific equipment.
Finally, Event Quests can be downloaded, and give you materials for specific equipment.
About Relics : Relics are randomly-generated equipment, which can only be obtained in Expeditions and Guild Quests. A maxed Guild Quest (lvl. 140) can give you equipment with better stats that anything you can craft. This is what Guild Quests are ultimately for.