We love Kefalonia and have been several times.
The beaches at Lassi and Lourdas are great for families as they have swimming areas where no motorised boats are allowed. Both beaches have sun loungers and parasols to hire by the day, and Lourdas certainly (can’t remember about Lassi) has pedalos to hire which is fun. The drive down to Lourdas is steep and twisty - Lassi is flatter and easy to reach.
The Argostoli harbour front is beautiful. Again, you can hire a pedalo here or walk across the bridge over the lagoon looking at the big shoals of fish and trying to spot turtles (we saw several). There are lots of places to eat along the harbour. Nice to eat in the central plaza of the town as well but a bit more pricey here.
Boat trips go from the harbour but will go from the other tourist spots too. Look for trips that offer swimming from the boat as that is great fun. You could also get a boat trip to one of the other islands - Ithaca, Lefkada or Zakynthos are all reasonably close.
Must-do otherwise - the underwater lake at Melissani and the caves at Drogorati. You can do the two in one day - it cost us €10 per adult to get a joint entry ticket for both.
The postcard beach on Kefalonia is called Myrtos - gorgeous white shingle- but be very careful there with little ones as it has a strong current.
Fiskardo, right at the north tip of the island, is the only town to have survived the 1953 earthquake, so the place to go to see the original look of the island. More or less everything else was rebuilt after the quake.
Be careful on Greek roundabouts! You’re meant to give way to traffic joining the roundabout but what we found was that drivers tended to make up the rules depending on how they felt at the time so aporoach with caution. There’s not many on Kefalonia anyway!