SureAI

So I done this quest a while ago and decided to explore around before advancing in the main quests. Now that I want to do the main quests I noticed that I don't have the next quest and The First Steps is still active with the 3 quest objective shown as complete. I tried going back to the room but neither Jespar nor Constantine is in the room and after searching the Sun Temple for Constantine, I found him reading a book and I can't talk to him. Was there something I missed in the quest?

Usually Jespar talks to you right after the investigation of Constantine and two new quests would start.
With the newer patches you shouldn't be able to leave the room at all. :/
Could you upload your current save game and a save game before the bug happened. The latter only if you still have it - it would help investigating what could have gone wrong. Otherwise we probably won't find out and can't fix the bug for sure.
But we probably get your current save game to work again.

Save games are located in this directory: C:\User\Documents\my games\skyrim\saves

Unfortunately, I don't have a save file that's close to where I was in that quest. The latest one I have is on a different playthrough where I met Jespar. Anyway, how would I upload the save file?

Ok, then just upload your current save game. Save games consist of two files: one .ess and one .skse file. Put both files together into an archive (.rar or .zip or .7z etc.), the forum software can't handle .skse files. Beneath the reply text field is a button called "attachments", simply click on it. and add the file.

Did I do this right?

Yes, and your save game is fine. Go into the Chronicum, Jespar is located down in the Archives. Just get close to him and he will continue talking to you, the quest will progress properly. The fast travel into the South Quarter might be buggy, but there is nothing we can do about it.

Alright, thanks for the help.