Dawes Arboretum near Newark is always worth a trip.

Not outdoor but the Franklin Park Conservatory has some wonderful palms, ferns and exotics etc in the butterfy house and Palm room. Also a collection of Bonsai and azaela.

If your in cincinnati, Glenwood Gardens is worth checking out. I would suggest waiting till june. The cincinati zoo also has some nice plantings, as does spring grove cemetary.
1 & 3 above are having bygl gatherings this season which would be a good time to visit.

There's the topiary garden in Columbus that is worth the stop. It's fashioned after a painting, the one that was hanging in the living room of the Cosby Show. I can't remember the name. Very interesting.