If you can spare the day a trip to Elmina to see the castle which was a major point in the slave trade route (but you’re only there for 3 days… )

Labadi beach is always popular.
for a quintessentially African experience should ride in a tro tro (it’s just a taxi / bus thing but you are really close to people and you’d be surprised how friendly (no not fresh) they can get.
Actually there was a scene in Last King of Scotland that depicts this.

Makola Market (give you a low view of sorts).

Villiages are not that far out past Kumasi but you aren’t there long enough for that.

If you want a really great club scene place OSU has a number of them and I totally recommend Fusion.
There’s a great Irish pub there (no I’m not kidding ) It’s called Ryans
Osu also has some great Chineese Restaurant (regal, dynasty) and one of the best Lebaneese joints (Club 400) but if you want a taste of the local food , go during the day time to Okaishie (it’s a teaming market place and you’ll find make shift restaurants (try Kelewele (fried Plantain) and Fufu and Banku (course you’re eating at your own risk )
good luck and let me know if you have any more questions

Oh and if you like Beer, you’ve to try Club and ABC beer (Ghana’s kinda famous for it’s beer).
And should buy some Golden Tree Chocolate (you’re after all in the birth place of most chocolates).