The peninsula (the land in the middle of the photograph below) is a protected nature zone and has pristine Mediterranean countryside, with exceptional views in all directions. There is an excellent circular route that you can follow by taking Ul. Ivana Meštrovića, which is the road to the left of the school. It goes up the hill and makes a bend into Ul. kralja Tomislava. An unmade road leads off from it at the end, passing through pine woods and olive groves and eventually reaching a tarred road. Turning right there takes you through the village of Rudina and down a long hill back into Stari Grad. You can, for a longer ride, turn left at the tarred road. It eventually reaches a Cold War fortification at the end. A more leisurely way to get there is by a taxi boat to Vaiana Beach, a beautiful cove where there is a seasonal restaurant. It is a nice walk back.
The field system here is the oldest in the world and is a World Heritage Site in consequence. Olives and grapes have been harvested there for 2,400 years. You can cycle (or walk) through it on traffic free roads. Some are unsurfaced.
This charming fishing village, called Little Venice because of its canal and bridges, is 7km from Nine Olives, on the other side of the Stari Grad plain. A quiet, paved road runs for 3km from there around a headland to the lovely town of Jelsa, where it is easy to get a bus or taxi to Stari Grad. There are nice places to swim. A relatively easy cycle circuit goes inland (on a quite tarred road that used to be the main road but is now bypassed) from Jelsa through the villages of Vrbanj and Dol and back to Stari Grad.