Ernakulam, the urban counterpart to Kochi on the mainland, doesn't have much to explore. But it's got plenty of good Indian eating places, in all price ranges. Certainly a number of places to eat in Kochi as well, but they're almost all aimed at the tourist market and in my experience a bit disappointing food-wise. Besides, the brief ferry ride from Kochi to Ernakulam and back is great.
The backwaters tours normally start in Alleppey, and it's a good place to stroll around for a day.