Both states are really all about the nature tourism as that's what brings you all to Borneo at all. While in Sarawak I visited both Bako (pronounced barkoo) and Gunung Mulu NP which you have to either hike in via the head hunters trail or fly into. You can't drive to Mulu which is actually pretty cool.
Chin and I headed to Bako from Kuching. Its only an hour on the local bus and easy to get to as its bus no 1 and Bako is the last stop so you can't really go wrong. Once you get off the bus you head inside, pay your park entrance fee and the boat fee and don your life vest and into the boat for the 25 min trip to the park. Be prepared to get your feet wet! We went out on low tide and had to walk over 500 metres of mud flat to get to the main entrance. Despite the coming and going - the paths around the park are litterly jungle tracks and there is a lot of water about. This is Borneo remember - so again - shoes you don't mind getting wet.
Our main goal of the day was to see the proboscis monkey but sadly I think we were the only two people today to not see any, however we had a great day doing out best Lara Croft impersonations climbing over the roots and vines on the paths and trails.
I didn't even know there were Macaque's (common as dogs monkeys) in the park until I put my packet of crackers down on our table to eat for lunch. Litterally maybe 15 seconds went by, which was enough time for me to zip up my bag and grab my water bottle and a Macaque had bolted in and swiped my crackers. OMG - they are so quick! We were pissing ourselves of course and the park kindly donated me a new bag of crackers to replace those which were now monkey food.
We ventured out on the trail that other hikers had said they saw the proboscis monkeys but sadly no go. Out of time we boarded the boat and headed back to the bus stop. Knackered completely but we had a great day and easily broached the English-Chinese communication barrier.
I had prebooked three tours which was a few different caves and the Garden of Eden walk. Most of the caves you can't enter without a park guide and two of my tours were only half day tours, with the Garden of Eden being a full day. The half day tours are pretty great as to get to most of the caves you have to go upriver in a little local boat (often having to get out and push) and then your caves are varying degrees of difficulty. The park staff are very knowledgeable and won't let you do anything outside your comfort zone anyway. There are quite a few steps (a couple of hundred in and out of most of the caves) and they can get a bit slippery but unless you pick a tour that is "adventure caving" you should be fine.
The guides are great and explained the evolution of the caves really well and in easy to understand terms. The actual caves are pretty amazing and all have different features and attractions. I really loved Mulu and spent my 4 days there either doing caving tours or walking the miles and miles of jungle paths. Unlike Bako - they were just paths through the jungle and very easy to navigate. It was pretty quiet and at one point I spent about an hour under a waterfall just myself on one of the more remote paths and only passed one other hiker that day, so very serene indeed.