Here I stayed at the White Rabbit Hostel. It had just opened up a day or two prior (they close during the off season - winter) so there weren't many people there. It was very clean and quiet.
If you can, I highly recommend checking out the Blue & Green Caves. You can also just rent a kayak and check out Hvar that way, though there are trails that wrap around most of the island if you prefer to stay on land.
I also went up to the top of Fortica, a fortress that offers lovely views from the top of Hvar Town. You can hike up there from Hvar Town or you can drive up to the parking lot. It does cost money to get into Fortica, though you do get nice views from outside so if you're on a budget, you can skip entrance into the fortress itself and enjoy it from outside.
On the second day, I walked around the island to Robinson's Beach. I had the whole place to myself. It's an absolutely gorgeous white rocked beach. There is also a restaurant here.