Water Taxis link the main harbour in Symi with several of the bays around the island, and in addition there are boats from Pedi bay to some of the other bays too.
The boats tend to run "outbound" 2 or 3 times in the morning and then "return" 2 or 3 times in the afternoon. Between times, they don't run at all and just tie up in the bay. It's a good idea not to wait for the last boat back, because if it's full, who knows if another will come and you could be a bit stuck. I've read some "horror stories" in the Symi Visitor site. While we were there there were no such problems, everything was as smooth as clockwork.
There is ONE bus on Symi. It runs from the harbour (Yialos), through the old town (Chorio) and down to Pedi Bay. It leaves Yialos on the hour, and returns from Pedi on the half hour. The trip takes about 15 minutes and costs 70 cents. We never used the bus - we walked everywhere. What can be better then climbing the steps into Chorio? The views are magnificent, the sights & sounds on the way up are not to be missed, and after all we need the exercise. Still, the bus is there if you want it.
You can only get to Symi by boat - there is no airport on the island. We travelled from Rhodes, and there are several boats to Symi from Rhodes each day. Note that there are two types of boat - daytrip boats and the ferry.
Now, these are the same boats, the only difference is the schedule that they sail under. Due to harbour restrictions the daytrip boats are not supposed to take passangers one way only, and are not supposed to allow luggage onboard. However this seems not to be true, so the best idea is just to ask.
As it happens we travelled on the "ferry" schedule, so we were OK anyway. The tickets are 12 euro one way or 20 euro for a return. We went on the catermeran Panormitis and returned on the hydrofoil Aegli. In addition the ANES company operates a second catermeran Symi II. All of these boats operate as both ferries and daytrip boats on different schedules.
The Aegli and Symi II bot go from Mandraki Harbour in Rhodes, Panormitis goes from the Kolonna.
Up to date timetables and costs can be found on the Symi Visitor website.