Inside the archaeological site, you walk. Total distances are not small — the round trip from the visitor centre to the Monastery is about 8 km, more if you add side trips — and there is virtually no shade.
Bedouin offer horse rides for the first stretch from the visitor centre to the Siq entrance (technically included in the ticket; a tip is expected). Donkeys, camels, and small horse carriages work the routes inside the site. Take a horse-cart through the Siq only if you have to — at canter the ride is teeth-rattling, and the carts crowd a narrow space.
Wadi Musa town is small and walkable, but it climbs steeply away from the site entrance — saving a uphill walk at the end of a long day is worth a taxi.