Fastest way will be to group up using PF and do N Than fates. It shouldn't take very long, especially right now that there's still some people lv'ing the new jobs.
Second best option would be dungeons but I'm not sure how bad the DPS queue times are right now at that lv. As for your dungeon hung up, I'd recommend you just get over it and if you can't maybe you should move on to other games. There's a lot of dungeons and boss fights ahead of you and most people in DF are in there for the exp or the bonus so you're gonna get grouped with a lot of speed runners. You can try stating that you're new and would like to experience the dungeon blind, but most people are overgeared for the 2.0 dungeons so even if they oblige you you might not get the experience you are looking for. If you really really don't want to watch videos or speed run youe only option is gonna be to find friends with the patience to do all dungeons like that for you or people new to the game who want to run it like that. Also DO NOT DF THE FIRST 2 LV 50 STORY DUNGEONS. They are long 8 man instances filled with cutscenes and are also in Main Scenario Roulette so you'll likely be grouped with people who do not want to be there and want to get it over with ASAP. Use PF or ask your FC to help you with a cutscene run, if you DF it you'll likely be locked out of most bosses while watching the cutscenes.
You are probably going to be out of luck on exploring a dungeon like you want. Even after your first run you will want it to go as fast as everyone else. The dungeons are 90% trash, and that is not fun for people to take their time on over and over again. My advice is to watch a quick guide to the dungeons so you know what the actual mechanics are for bosses. Then just run through the dungeon with them and if you want to explore you are free to run around in the cleared dungeon after the clear. Most of them are pretty linear though so there is not really much to explore.
Thanks. This gives me something to think about going forward.