The #-# type of quests are the government quests. 1-1 ~ 4-X (I forgot how many) is the Episode I storyline, 5-1~8-3 is the Episode 2 storyline, 9-1~9-8 is the Episode 4 storyline. To access the quests, 1-1, 5-1, 9-1 are available from the start, but the rest you have to unlock by doing the preceding quests. For 6-3 you need to finish all of 5-x, then 6-1 and 6-2. Hopefully that's clear enough.
Also, the quests are unlocked only on the difficulty you play it in, so if you finished all of 1-X, 2-X, 3-X, 4-X in Normal, they won't be unlocked in Ultimate. Government quests also have quest rewards if you talk to Irene after you finish each of 1-X, 2-X, etc. Most are junk (though sort of useful for newbies) but there are a few decent items on there, mostly in Ultimate only though.