I joined a private club in Powell a couple of year ago, but I would recommend the following courses:
Longaberger - must play once a year to enjoy the atmosphere, well maintained and a great staff.
Cook's Creek - fun course, just don't play it after a heavy rain as it is in a flood plane and can get messy.
Darby Creek - one of the best courses as mentioned above, lay out is fun and staff is great. Play early of later, though.
Clover Valley & Links at Echo Springs - kind of out of the way, but still fun - make sure you hit it straight or have plenty of balls with you as the fescue off of the fairways at Clover Valley is pretty nasty.
Cumberland Trail - some of the best public greens in Central Ohio, can get backed up on the weekends.
West Chester - again nice layout, well maintained go in the afternoon to avoid the back up.
Golf Club of Dublin - fun to play for a change of pace, it's a links style course, but way
OVERPRICED!!
Royal American - good course, play in the evenings to get a better rate.
Oakhaven - extremely different nines, front is wide open, back can get really tight. (Plus the pro is my buddy).
Places to avoid like the plague:
Blackhawk - overplayed and not well maintained - greens are rock hard and your feet will hurt like hell.
Shamrock - bring your hard hat, really tight and way too many golfers on the course at one time. I know people who have been hit by balls on the course.
The Phoenix - its build on a old garbage dump need I say more? God only knows what carcinogens are in the soil.
Bent Tree & New Albany Links - they have way too high opinion of themselves in terms of the course and staff members.
Private Courses to play, if you can get on:
Scioto
Muirfield Village
Tartan Fields
The Golf Club
Double Eagle
Wedgewood
Brookside
Pinnacle - after the first three holes its an awesome course - golf porn.
I will probably get hammered for this, but I played OSU a week before the Nationwide event and was not overly impressed. A lot of the holes seemed the same to me. Its crazy long from the Buckeye tees, though.
Good Luck and play well!!