Hello and welcome to the forum.
I'm sure you will get plenty of responses to your questions. These are mine:
This Google Earth image might help you understand the geography of the place. It shows the main harbour area (Yialos) and, towards the bottom right, "The Village" (Chorio).
As you can see, nothing is very far from anywhere else!
The ferry will dock at the 'point' of the main (lower) part of the harbour, just above the centre of the image, at the opposite side to where the large ship is moored.
If you pre-book your accommodation the property owner or agent will arrange to meet you where the ferry docks and take you to the accommodation. It may be a short stroll along the harbour front or you will be taken by car or minibus right to the door so there will be no transport problems when you first arrive. Virtually all of the accommodation is in Yialos, around the harbour, or in Chorio which is on the hill overlooking the harbour. There is a hotel in Pedi, a small bay (not shown on the Google Earth image) to the south of Yialos.
There is one bus which leaves Yialos, around the area where the big ship is moored on the Google Earth image, every hour to take you up the hill to Chorio and then down the neighbouring valley to Pedi Bay, from where it returns to Yialos (via Chorio) on the half-hour. There are also taxis on the island and the main taxi rank in Yialos is on the south side of the harbour, not far from the bus terminus. So getting around Symi is no problem, regardless of your age. Trust me, I know
I don't have any suggestions for cheap accommodation though others may offer some tips. There aren't many hotels on the island, most of the tourist accommodation tends to be in studios or apartments.
Regarding your next stop after Symi you are constrained by the ferry itineraries of course. Most of the timetables haven't been published yet but the fast catamarans heading north can get you first to Kos which is a bit busy for my tastes and then on to Kalymnos and Leros, both of which are worth a visit. On certain days (Friday I think - could be wrong) it's possible to get to Tilos on the Bluestar ferry so that might be worth considering. In the opposite direction, Kastellorizo is a possibility though, again, probably only one day a week and since the timetables haven't been published yet it's too early to be sure it will fit in your itinerary at all.