There are 2 main types of quests in PSOBB, on one side you have the Main Story quests which you can take from the Mayor's Room in the Pioneer 2 City (The Episode 2 ones are taken from the Laboratory room)
The other kind of quest you can take from the Hunter's Guild, and these are separated between online quests which you can partake with 1 or more member or via single player in the Side Story category, offering in many cases a totally different story line from the Main Story ones.
You can find these quests in the Mayor's Room by speaking to an NPC called Momoka, in addition to unlocking new hunting areas, we recommend completing these quests to ensure your character is leveling up in the most efficient way to tackle the harder difficulties.
About Episode 2 and Episode 4
You can start the Episode 2 Main Story quests by speaking to Nol in the Laboratory area of the Episode 2 Pioneer 2. Likewise for Episode 4 just speak to Momoka like you did previously in Episode 1.
Theres also another category within the Hunter's Guild which is available only if you start a new "Solo Mode" game called Side Story, Side Story are a series of quests containing a Storyline on their own, on top of being easier to hunt in this mode, they can offer you greater insight within the PSO world and it's characters.
About Single Player Mode
By selecting "Solo Mode" in the lobby while you create a room you will be granted access to the Side Story category in the Hunter's Guild counter, and for example you can take a quest which is a small tutorial that will explain the basics while you follow a small story for the search of a lost hunter in the Forest. Or helping a stranded Android in the caves while she explains all the details about your Mag unit.
The stories of these quests get more and more interesting as you try more! check them out!