The slimes you collect are distinct from the ones that you'll see bouncing around who will attack you. The collectable slimes are docile and can be found attached to walls or hiding in out-of-the-way places.
While there should be a slime right after the tutorial, it appears it does not spawn in until you progress a little. Proceed through the port and at the far end of the castle a long scene will play.
Keep heading in this direction (most of the area is blocked off so you can't really get lost) and when you approach the doorway Valy will talk to you. Now go through the doorway. Around this time a health bar will appear in the top-left.
The new area will be Temple Field. Head forwards and around to the right when you can. Note the trees here:
The first slime is up against the wall behind the trees.
Simply walk near it to collect it. Note the chirping noise it makes and the icon in the top-right of the screen - be aware of these as you play so you can collect the others.