Edinburgh buses are brilliant, but the town is a nightmare at the moment because of all the roadworks going on for the bastarding trams. Still, buses are a lot better than bothering with a car. Edinburgh is so small that nowhere really takes more than half an hour on a bus.
If you are out near Corstorphine, that is next to the zoo and so thats a definite for a day out. Also there is the botanic gardens, down Stockbridge/Cannonmills way, which you can spend a whole day at in nice weather. And its free. Stockbridge itself is very nice to just walk round.
There is also Princes Street gardens, obviously, and walks round the Royal Mile and the Castle. And Portobello Beach, which if you are staying out at Corstorphine I'd bring the car as its the very east end of the city. Also just wandering down at the Shore in Leith is lovely in good weather, and there are loads of nice bars and restaurants.
The Meadows is the biggest park in the city centre, its huge, and is always busy when the sun is out.
Maybe take a wee trip down to North Berwick one day, its really lovely and there is loads to do. Or the safari park, but I'd say yor DD is a bit too young to care!
Have a nice time