I don't want to hijack your holiday Sue but how about this for an idea for your exploration?...
"cheat" and take the bus up to Chorio. As well as saving your legs, the Symi bus is an interesting fun experience. As Andy mentioned, it leaves (roughly) on the hour every hour up the hill to Chorio and on to Pedi, returning from Pedi on the half-hour, you can get off where ever you like. The bus stop in Yialos can be seen on the picture in the Iapetos thread, on the left hand side half way up. You can see it's a short walk from your hotel.
Get off the bus right at the top of the ridge, by the ruined windmills. Then walk from there up the ridge track past the windmills. The views from up there are spectacular and not to be missed. Your camera will get a LOT of use!
You can see back down to the harbour.
Right over the whole of the village (Chorio).
And down to the fertile Pedi valley and Pedi bay beyond.
Then, before you completely fill your camera's memory card, retrace your steps down to the road where you got off the bus and take the small road which takes you down into the centre of Chorio, bringing you out right at the top of the Kali Strata just by the village square and the two kafenions that SueO mentioned. I strongly recommend you sit yourself down at one of them and have a cooling drink of your choice before exploring some more...
I echo other poster's comments about the delights of wandering the maze of tiny lanes through the village. It is one of my favourite things to do on Symi and, as others have said, it's still possible to find new unexplored areas even after many visits to the island. Needless to say, the photo opportunities are endless.
SueO's suggestion of following the published guide is excellent. The pictures are very helpful and if you follow the instructions you are unlikely to get lost but even if you do you're not really "lost", just a bit unsure and if you head downhill you'll soon get to somewhere you recognise. It's honestly not at all scary!
Assuming the guide works you will end up on the "Donkey Track" out of the top of the village from where you get even more stunning views. You can, if you wish, continue along the track for a while to Agios Paraskevi church which is worth a visit if you feel up to it or turn back downhill and enjoy yourself wandering aimlessly through the lanes and hidden places of the upper village. Eventually you'll end up back at the village square and the welcome kafenions. From there (after another refreshing cold drink or two) you can wander down the Kalistrata (more pictures!) back to Yialos.