In terms of furry characters...
Rafiki can be met from about 9:30-10:30am in front of the entrance to AK. He is pretty much the only Lion King character to meet. You can also see him at Rafikis Planet Watch in the AK.

Baloo and King Louie have regular scheduled meets in the AK as do Dug and Russell from UP.

Marie can be met in Town Square every day.

White a Rabbit meets with Alice near the Tea Cups.

Meeko does not meet regularly anymore. However he will sometimes turn up in DHS in the later afternoon on Streets of America unannounced.

You may catch BrerBear, Brer Rabbit and some of the Country Bears during the Frontierland Hoedown each afternoon in Frontierland. They don't do meets but you can dance with them.

Beast can only be met at BOG for dinner.

There's Duffy in Epcot too.

Hope this helps.