Getting around the island isn't the easiest thing to do without your own transportation. Colectivos run hourly along the road in both directions; it's COP$2500 for a ride of any distance. As there are such big gaps between buses, locals may stop and offer you a ride.
Taxis are hard to come by and are quite expensive compared to those on the mainland.
The best way to get about is to hire a scooter (from COP$70,000 per day) or a golf buggy (COP$130,000 for eight hours). This can be done at countless places around the island, including Providencia Tours, and your hotel will easily be able to arrange it for you. Note that local road signs use the English names (South West Bay, Fresh Water Bay, Manchaneel Bay), rather than the far more colloquially used Spanish ones, which can be confusing. The island's only gas station is in Santa Isabel.