Hi 42levels
I'm not sure I'll be much help but I travel to the Philippines every year as I'm a keen scuba diver and the diving there is second to none.
When in Manila I stay at the Shangri La or Peninsula so I'm not sure about hotels in your budget, but I can recommend some great restaurants in Makati if interested...

Anyway this year we wanted to see more of the Philippines and we went up to Sagada and Banaue - it is truly stunning and very rewarding but a royal PITA to get to. We hired a driver and guide to get us there and show us about, and stayed in the Banaue Hotel which is an experience in itself. It's a good solid 10 hour drive each way. Flying is not really possible. There are buses that leave Manila for Banaue and then on to Sagada. The buses often travel through the night to get there, and stop several times.

Palawan is my favourite place in the Philippines. The landscape is stunning and the diving is wonderful. We always stay near a town called El Nido in Bacuit Bay. There are reasonably priced hotels in the town, or you may want to go further south to Puerta Princesa, the largest town on the island, but I haven't been there and can't comment. I stay in a place called Minoloc, part of El Nido Resorts, but the prices might put you off if on a budget.
Flights to El Nido can be pricey - we use Island Transvoyager. You might be better flying to Puerto Princesa on PAL.

When we went to Bohol, we did an overnighter. Left Manila early morning and stayed at a hotel called the Peacock Garden. Not on the beach, out the way in the hills. The next morning we hired a guide and driver to do the main tourist route - Chocolate Hills, Tarsiers etc. again, its gorgeous. We flew back later that afternoon.
Tagbilaran airport is a treat - very small. The departure gate singers are hilarious. Ask for an upgrade at checkin. We were given one for about $12.

A few bits of info. Probably not very helpful but any questions do ask or pm me.
Cheers.