I'm surprised that in order to learn about fusing, you have to enter the velvet room manually.  It would have made more sense to "lead" you after clearing your first dungeon floor into the velvet room using some dialog.
 
Short day, just cleared out the first floor, got out and saved.  Did my best Vinny impersonation by missing with my sword.  Also, wtf is Yukari's weapon?  Ah, bow and arrow, looks way too big.
 
It is kinda weird you have to go into the persona menu for the other characters to do analyze, would have made more sense in the main radial menu.
 
Looks like next up is social linking!