Kuching town itself is quite small and can mostly be managed on foot, but for getting to schools, attractions nd malls beyond the town centre, you may utilize the following.
1. Grab ride-share
Download the app on arrival.
2. Taxi
The small red taxi's are largely unmetered and you will need to negotiate the price. Some hotels use the big blue or gold taxi's which are metered and more comfortable, although a little more expensive.
3. Bus
Fares are cheap and bus timetables can be found at the Tourist information Center near the waterfront. You can also take the bus to outer lying tourist attractions such as Semengoh.
4. River taxi
For a few ringgit you can utilise the small water craft that ply their trade on the rive and you can be on the other side in minutes. There are several crossing points along the waterfront and crossings are frequent. You can also negotiate a cos for a group to go further upstream or downstream.