Hello and welcome
The cruise ship visits the island for only a few hours so your options are quite limited.
Since virtually all of the island's population live in the main town where your ship docks or in the village above there are few things to visit outside that small area. The only significant site you might want to see is the Panormitis monastery which is down at the south of the island. You should easily be able to get there and back in the time you have available and it's certainly worth a visit.
I suggest you contact Symi Tours travel agency on +30 22460 71307 or +30 22460 71689 Their website is
http://www.symitours.com They run trips to Panormitis in their mini-bus and if you can get a number of your fellow passengers interested they may put on a special trip for you. Give them a call and explain what you want.
Taxis are based on the harbour front at the opposite side to where your ship docks. Their phone numbers are listed on the SymiBest site
http://www.symibest.gr/services/ The prices are reasonable and whilst they don't offer guided tours they will happily drive you to Panormitis and back. The drive is interesting and you will see most of the island as you go.