I've run into a bit of a snag. Usually, I can average 2 levels an hour just running quests, but when I hit about halfway into level 18, they stopped. Cold. I couldn't find anymore quests anywhere. So I ground my way to level 19. One quest was all I could find. I've been running back and forth between the level 15-17 area, and the level 18-20 area, with no luck. Is the game going to force me to grind on mobs to reach level 20?

Use your scouter to find quests.
You need the quest chip which you get for free during the starter quests.
Press T then the 2nd button on right panel.
Blue/red dots should appear on your quest map, thats where you can find quests.

also if all else fails, go back to somewhat lower quests to try and make a level or move on to another map and see if ya can find some quests there

Also, do not forget to check the starter dungeons. Some quests could be in there!