Looks like you are getting some good responses there Lisa and it is lovely to get a "right now" view of Symi - thanks Andy.
Regarding walking to beaches, Gemma has more or less covered it for you. As with so many questions about Symi, the correct answer is: "It all depends". Not very helpful, I know
Everyone is different and one person's 'short walk' could be someone else's 'strenuous hike' but there are always options so see how you feel when you get there.
You haven't mentioned where you're staying on the island but if you are around the harbour area in Yialos then, as Gemma says, there is Nos beach just round the headland and the walk there is completely flat so shouldn't trouble anyone. You can continue round the road to Emborio (or Nimborio - choose your spelling). The walk isn't as interesting as the one that Gemma suggests but it is almost flat the whole way. Again, a different option.
If you are staying in the village (Chorio) it is an easy downhill stroll to Pedi (you can get the bus back if you want) and the beaches there are fine with two good tavernas to choose from. Alternatively you can walk from Pedi over to Ag. Nikolas as Gemma suggests, it's a nice walk. Just one thing to be aware of though: The last stretch of the walk is down steps cut into the rock and, whilst it is perfectly safe, it might not be suitable for anyone that suffers from vertigo.
A longer walk from Pedi round the other side of the bay will bring you to Ag. Marina.
There are options again here. There are taxi boats in Pedi that will take you to Ag. Nikolas and Marina to save the walk or you could choose to walk there and get the boat back, or vice-versa.
More distant beaches are best visited using taxi boats. In fact, St George's beach is only accessible by boat. Many people love the exhilarating taxi boat ride to the beaches (those boats go fast!) and make the boat trips a big part of the holiday.
So many options!