Hi benjoe ~ yes, you can walk from Koukounaries (bus stop 23) through the forest to the Mandraki beaches, Elias and Xerxes, or from bus stop 25 up the road by the Golden Beach hotel to Aghia Eleni for fabulous sunsets and on to Hidden beach. Also, from BS 26 (end of line) you can access Ambelakia, Big Banana and Little Banana. All of these are fairly easy strolls.
Depending on how much you like walking, you can also walk up from Troulos crossroads (BS 18) and a fairly long-ish uphill walk will take you to Asselinos, Micros Asselinos,Ligaries and Kechria beaches. I've only ever been to these by jeep - walkers advise taking water and not trying it when it's too hot!! The views are amazing from all of the back roads. If you can, pick up a free map from the airport arrivals hall & you'll see them marked on. Woody has done a section on here called Route Maps - very useful. (Or use the Google map at the top of this page)
Thanks Dave - forgot about that one, though never done it myself!
There are, of course, beaches you can visit by going on a boat trip - eg Lalaria, Kastro & Tsougria. For a good overview of the island, if you've never been before, I would recommend a day doing this. However - book at the old port and NOT with your tour rep! Twice the price for the same trip 😱