I would definitely go to Bako national park...it's one of the oldest forests on the planet and there are even noticeable microclimates in ascending the hills. Kubah national park is also within easy reach though I didn't get there. Besides the rainforests, the meet-the-natives experience is a must, though the better longhouse trips inland require at least an overnight (and some outlay). This can create a dilemma if in Kuching on limited time because one of the attractions of the city is how cheap the 4-star and above accoms in town can be if you time it right. The riverfront and all the cafes and restaurants in the colonial buildings can be an easy walk from some hotels.

The hurried alternative to the longhouse trips is the Sarawak cultural village, which is a bit touristy but thankfully had just about the right number of visitors (not too many, not too few) when I went.