Kutaisi is the capital of Imereti, and Imeretian cuisine has its own distinct character within Georgian cooking. Imereti is cheese country — the mild, fresh Imeruli cheese appears in khachapuri (the bread-cheese boat), in salads, and on its own. The Imeretian khachapuri is round and flat, distinct from the Adjaric boat-shaped version more familiar to tourists. You will eat better khachapuri here than almost anywhere else in Georgia.
Walnuts appear everywhere — in pkhali (compressed vegetable dishes), in sauces, in churchkhela. Local wine from Imereti's Rkatsiteli and Tsolikouri grapes is light and dry by Georgian standards and pairs well with the food.
The best eating is often in small family-run restaurants (often called "guests house" or just named for the proprietor) rather than the tourist-facing places near the central square. Ask at your guesthouse for current recommendations.