There are no roads to Bako National Park. The only way to get in is by taking a boat from Bako Bazaar (about 20 minutes from the center of Kuching). As of 2017, boat rides are pooled and arranged for you so you don't have to worry about finding other people to split the cost with. The ride will cost RM20 and you will be placed with other passengers. The shared boats all depart for Bako National Park in a single wave around 8AM-9AM, so try to arrive early.
Odds are good you will arrive during a low tide. Be prepared to disembark from your boat in the water and wade to shore. Wearing shorts and shoes you can remove easily is a good idea. Tourists storming the beach is an essential part of the Bako experience.
Some people try to do Bako as a day trip. The return boats leave relatively early in the afternoon so if you do try to return the same day you'll only have a couple of hours to explore the park. There are plenty of cabins and rooms for rent in Bako which can be booked from the Sarawak Government for fairly low rates by Western standards. The booking and payment system can be a little glitchy so it may take you a couple of tries.
There are a number of trails you can access from the visitor center which acts as a hub. Most of the trails are long loops. If you are doing a day trip you may only be able to fit in a quick hike to one of the closer beaches.
Wildlife is one of Bako's biggest draws. The Proboscis Monkeys may be the biggest draw in the park and you won't have to try very hard to find them. There are usual several hanging around near the park entrance and they are fairly ambivalent to humans being nearby. You'll spot a number along the jungle trails as well. The Bearded Pigs are a constant sight near the visitor center and cabins as they rummage for trash. You should also be careful setting down bags as there are absurdly hostile macaques waiting around to snatch anything they think may have food.
One of the perks of staying overnight in Bako is the guided night hike. The rangers can point out all manner of insects and nocturnal animals, including the poisonous pit viper which lurks in trees completely still waiting for prey to happen by.
I opted to hike to Telok Pandan Kecil the morning before I returned to Kuching. This was a moderately-long hike and it took several hours to reach the beach. Don't underestimate the time as you will making some ascents in very hot and humid conditions. The trail takes you north of the visitor center and up across some highlands that are much more arid than the south-bound trails into the jungle.
Telok Pandan Kecil is an excellent choice if you're trying to decide which trail to take in Bako and have the time. You could swim at the beach, though there are reports of spiny catfish in the area so take caution.