Hi,
The problem you currently face is a bug which will be fixed in the next patch along with several other fixes.
For temporary fix : reload to the step where the quest says 'Kill 3 Adult slimes' - that should let you progress The story. This bug is extremely rare and random (I've tested it dozens of times and never triggered it once), so the next patch should eliminate it completely.
Katarina Town unlocks after you finish the Slimy Business quest, so you'll get access once this is resolved.
Thanks for the comment.