True Vault Hunter mode is essentially the next level of difficulty up from normal. You play through the game just the same, but the enemies and loot are leveled higher than normal. All quests are tracked separately for TVHM and has no interaction with normal whatsoever. Your normal mode progress including where you left off is saved for next time you come back to it. Everything else such as badass ranks and loot will carry over bextween both modes.
You can switch between the two modes freely from the first menu screen. When you load a character after unlocking TVHM for it, you will have the option to choose between normal and TVHM.
Note that unlocking TVHM for one character will not unlock it for any others.
I read that I shouldn't complete any side quests until level 50, but
how does that work since don't they drop a gun that is the level the
quest is (e.g. a level 40 quest always drops a level 40 weapon even if
you're level 60), so it will be the same either way?
This is true, a level 40 quest will drop a level 40 weapon. The thing is, the level of the quest is determined by which level you are when you accept the quest. If you accept a quest at level 40 it will reward you with a level 40 item. Instead, you can hold off on accepting the quest until level 60, and the quest will also be level 60 and reward you with a higher level item instead. The reason you want to wait as long as possible is that most quest rewards are unique meaning they can only be obtained through completeing that quest