In reply to RBonney: I'd say almost certainly yes. It would be worth buying a walker's map (e.g. "Kompass") from Stanfords. Not too expensive and it shows all the paths.
From my recollection, Cala Magraner is in the shade early in the day and receives the sun later; it may get too hot in the direct sun in June.
You ask about other places we think are suitable. I'm not clear where you'll be staying - one venue or more than one? One obvious place accessible by public transport is Sa Gubia (range of grades, range of aspects so you can avoid direct sun if you need to). Fast Palma to Soller bus passes the restaurant "Can Penassa" or there's a local bus to Bunyola village or there's the veteran train also to Bunyola.
The Valledemossa crags are a tolerable walk from Valledomossa village or shorter from the first bus stop after the village.
Cala Serena (by Cala Ferrera near Cala D'Or) is accessible by bus; a massive DWS venue but the sea and requirement to rappel may be intimidating for novices.
I believe Porto Christo town is accessible by bus and it is a short walk to Cova del Diabolo and the other venue.
There is a resort just south of Porto Christo called Cala Romantica which has buses. I would expect it to be about a 40 minute walk from there to Cala Barques.
Bus times are on "TIB". Be aware that most of the holiday resort services don't run at all at this time of year so try and find the map of summer services. There a summer map on there as well as the main winter one.