The best way to get around Bangka is to rent a motorbike or car in Pangkalpinang. Roads are in reasonable condition — better than much of provincial Indonesia — and the island is compact enough that you can reach the northern beaches from the capital in about an hour. Rental agencies are straightforward and car rental with a driver is a common option if you'd rather not navigate yourself.
There is no reliable public transport connecting the beaches and tourist areas. Ojeks (motorbike taxis) are available in town, and you can arrange transport through your hotel, but for any kind of independent itinerary — multiple beaches, the mining lake, the forest reserve — having your own wheels makes a real difference. Fuel is cheap and the main roads are signed clearly enough.