Let me try to give you specific answers to each of your questions.
1. On Santorini there are no taxis at the tender pier and you must first get up the hill to the town of Fira by either taking a donkey or using the cable car. It is also possible to walk, but you have lots of steps and would be also walking along the donkey path which is full of a lot of donkey poo. Once you get up to Fira, you can only explore this town on foot. There are taxis that can take you anywhere on the island and there is also a public bus line that can take you a lot cheaper to Oia. Santorini does have other things on the island such as some nice beaches (we like Red Beach) and a few interesting wineries. To explore the island away from Fira and Oia it is best to rent a car.
2. Mykonos has limited taxis and the situation depends on whether you dock (about 1/2 mile outside of town) or tender into Mykonos Town. The town is small and can only be explored on foot and has some good shopping and nice cafes. Mykonos is a great beach island and you could certainly take the public bus from the bus station (near the windmills) over to Paradise Beach and have a nice beach day.