Ferry services connect from Rassada Port in Phuket Town to Ko Phi Phi and to Krabi Province on the mainland twice a day, taking 90 to 120 minutes and costing 350/650 baht one-way/return, for each leg. It's usually a pleasant ride, but can be rather bumpy when it's windy.
From the harbour, you could avoid the minibuses and take a 10 baht songthaew to Phuket Town. If it doesn't show up at the bus stop right outside the terminal, you'll have to walk past the gate outside the harbour and along the road, turning left at the first T-junction, about 600 m, then on the main road you can catch a cheap songthaew. There's a picture of all the routes posted just outside the terminal near the bus stop inside the complex. Last one leaves at 19:00.
There are speedboats from/to Ko Racha (45 minutes), Phi Phi (1 - 1.5 hours), the Similan Islands (about 3 hours) and other islands. Boats and yachts can be chartered all year from Phuket at Chalong Bay, Rawai Beach, the Boot Lagoon, the Yacht Haven and Royal Phuket Marina. Boats from Phi Phi and Phang Nga can be found by visiting the local beaches. A search for Phuket speedboat charters will turn up many companies providing inter-island charters and services.
Prices for transfers on board a tour speedboat are typically between 1,500 to 2,500 baht depending on destination. Phi Phi speedboat transfers (no tour) are provided by the Zeavola Resort, which has dedicated speedboats for Phi Phi transfers. Most companies doing Phi Phi speedboat tours will not accommodate transfers that include baggage due to space limitations.
Prices for speedboat charters to/from Phuket range from 3,000 baht to 60,000 baht+ depending on distance/size of boat:
It's possible to visit Phuket by cruise ship. For cruises from Singapore, try Star Cruises