Next to tricycles (the three wheel motorized taxis), scooter and motorcycle is probably the most popular way to explore Palawan amongst foreign tourists. The island has miles of scenic roads with little traffic, lending itself well to exploration on two wheels at your own pace
Where to rent: there are lots of bike rental shops in both Puerto Princesa and El Nido in their main commercial areas. I've listed some popular ones at the bottom
In Puerto Princesa, if you go to Rizal Avenue, which is very close to the airport and hotels, you will find many rental shops.
In El Nido, the port town has most of the bike rental shops
In Port Barton, I've heard that there is a small number of bike rental shops but I don't personally know of any
In Coron, which is a separate island from the main Palawan island, there are a number of rental shops in the port town of Coron
Price: the price ranges from 300 pesos ($6 USD) per day to 800 pesos ($16) per day. On average, El Nido bikes are around 500 pesos per day ($10 USD) and Puerto Princesa bikes are around 400 pesos ($8 USD) per day. Of course, you will need to negotiate to get it down to this price level as they will always quote you closer to 800 pesos. Cash is preferred but some rental shops also accept international credit cards like Visa and MasterCard
What you need in order to rent: aside from a deposit, all bike rental shops will ask you to leave your passport with them. This is a very standard practice, not only in Philippines but elsewhere in Southeast Asia too like Thailand from my experience. I've not heard of any instances of rental shops losing or refusing to give your passport back. If you don't feel comfortable leaving your passport, you can always try to negotiate and see if they will let you leave your driver's license with them; some places will be find doing that and others will insist on passport.
Do you need a motorcycle license: legally you need a motorcycle license from your home country in order to rent a scooter and motorcycle. However, this is never enforced in Palawan; I've never come across a rental shop that demands to see your motorcycle license. Police most of the time turn a blind eye to it because it's just such a standard practice (and I suspect they get a cut of the profits from the rental shops). In the rare cases that you get stopped by a police, you will just need to pay them a 1000 peso "fine" and you'll be let go. Where this becomes a potential problem is if you get into an accident. If you don't have a motorcycle license, you will not be covered by insurance, so you're liable for any medical costs you incur as the result of the accident.
Other things to be aware of:
Most motorcycles are manual rather than automatic gear, so make sure you know how to ride those. If you only know automatic, rent a scooter instead as those are all automatic
A lot of bike rental shops do hotel drop offs where they will bring the scooters and motorcycles to your hotel for you in the morning of the day of your rental. Ask your rental shop for this option when you rent and can save you some hassle
Make sure you know when the rental shops close, as you don't want to come back after that time and not be able to get your passport back when there's no one around to receive you
I highly recommend reserving a bike ahead of time (either online or in person). During busy season the there may be a shortage of bikes
A short list of popular and/or well rated bike rental shops in Palawan: