I hope you are prepared for a big portion of cuteness overdose, as I have selected the best out of the best from what animals you can commonly find around Iceland
They are all over the place, probably more towards the South
They are mainly everywhere, but for a nice horse-riding experience, try Akureyri.
Check out more details on my post about "Exploring the Snaefellsness Peninsula"
For where to find them check out my post about " Where to find puffins"