Parking is a total disaster, in fact, it's hard to find a worse place to park a car in Europe. There are no underground parking, the center is one-way traffic and only one parking lot, on a large lawn 39.624861, 19.925139, which is the central square "Spianada" or "Esplanade".
We were with a child and finding a good playground on the island is also a trouble. There are two playgrounds in the center of Kerkyra, one near the green lawn (39.624722, 19.925444), not particularly good.
The second one is near the harbor (39.626417, 19.918306), but getting to it is a challenge, considering everything is parked.
It would not be easy to see everything in the center in one visit, we came several times. At first we walked along the waterfront. From the parking lot we went straight to the wonderful Garden of the People park (39.625222, 19.925389). For a long time the park was closed to the common people and was only accessible to the nobility, but now you can walk there freely. In front of the entrance to the park stands a beautiful, pink church (Holy Church of the Virgin Mary Mandrakina, XVIII).
There is also the City Palace, built between 1819 and 1824 as the residence of the High Commissioner of the island. The commissioner, by the way, was the ruler of the historical country Ionian Republic (wiki), which existed from 1816 to 1864.
The park itself is very nice.
The park offers a beautiful view of the so-called Faliraki beach with a cozy restaurant. However, you can't go down from the park and if you want to get there, you have to walk a bit.
Then we walked along the promenade to the other end of the central square.
From this spot there is a view of the other side of the fortress and an interesting temple, which is now a church.
There is also a beautiful Music Pavilion in the park, a place for local festivals.
Then we walked around the old town. Some photos were taken early in the morning, so the streets were empty. This is the very center and the most famous street of Liston - Eleftherias. It was built during the second French occupation in 1807-1814, and the Rue de Rivoli in Paris was taken as a model.
Great place to hang out at the restaurants. There is a large lawn next to it, so if you are with kids they can play there.
Jean-Paul Belmondo in Corfu.
The old town is quite large and you can walk around for a long time.
James Bond in Corfu (39.62524, 19.9225)
And the third time we visited the fortress. The main attractions of Kerkyra are the fortresses of Paleo-Frurio (XII-XVI cc.) and Neo-Frurio (XVII cc.). For proper light the first one should be visited in the morning, the second in the evening. We visited only Paleo-Frurio and can recommend it, as the fortress itself is impressive and the views are the best.
The fortress was founded in the 8th century by the Byzantines, later completed by the Venetians and the English, so there are quite a few styles intertwined here.
The fortress, by the way, is huge. We spent at least 30 minutes just to get to the top observation deck.
Views of the city.
And here is the New Fortress in the distance, built in 1576 by the Venetians to strengthen the defensive power. By the way, two huge fortresses at such a close distance, in the center of the city, is quite a unique phenomenon, I don't remember anything like it.
Getting there: parking at 39.624861, 19.925139 or 39.626583, 19.917722. If you cannot park there then the nearest parking lot is 39.626028, 19.913222. Taxis from the airport are licensed and quite expensive (https://www.corfu-airport.com/corfu-airport-taxi/). Airport bus schedule (https://www.corfu-airport.com/corfu-airport-bus/).