Patong itself is walkable — most things you need are within ten minutes of Beach Road. To leave town you have several options, all overpriced.
Songthaews (blue local pickup-truck buses) run a regular route between Phuket Town and Patong, costing 40 baht — they stop at the eastern edge of Bangla Road. Beyond that, tuk-tuks and taxis operate a tightly controlled cartel; trips to other beaches start at 500 baht and bargaining is limited.
The cheapest way to explore the island independently is to rent a scooter — daily rates from around 200 baht. Wear a helmet, carry an International Driving Permit (Thai police checkpoints will fine you without it), and be wary of the steep climb out of Patong toward Karon, which catches out many inexperienced riders every year.
Grab (the regional Uber equivalent) is available but the local taxi cartel has limited its operation — drivers won't always pick up in Patong. For airport runs, pre-book a private transfer for around 700-900 baht one-way.