Are you supposed to go directly to the lobby after completing a quest? Because it does it every single time when I do so. Also a side question. What is the best way to do the single player experience in BB? Because it doesn't save the quests or the stages you have completed. Thanks.

In terms of completion, do you mean that you had talked to the guild lady to receive your reward? If so, then it automatically returns you to
the lobby.

Also, data is saved via server side. You wouldn't have a character at all if it was not saving at all.
Just make sure that you are in the same ship / lobby to get the same data to be retrieved.

Why does it bring you to the lobby? That is so annoying. I don't enjoy creating a new party each time I do a quest.

The way the game client is coded, after a quest is loaded it's impossible to load another one in the same room (without breaking it). Therefore, it sends you back to the lobby so you can create another room. It's the way PSO/PSOBB has always worked.

Simlpy put, hardcoded.
Mess around with the data and crash pso, I had a fun time reinstalling.