The video above from Lowe's shows how to fix leaks in all three types of single-handle faucets. Basically, you're going to remove the handle, pop out the mechanism, and replace it or the surrounding seals. If you can spot which part is causing trouble, you can buy each of the pieces individually. Otherwise, you can buy a faucet repair kit for that type of faucet and replace everything at once. Neither option should be very expensive.
Plumbers in Australia have licensing requirements that differ from state to state but it is generally accepted a 4-year apprenticeship with a further minimum experience of 2 years (6 years total) and a further curricular requirement as a benchmark for licensing. Licensed plumbers are also expected to maintain minimum relevant training requirements to maintain their plumbing license