High hiking shoes and walking sticks. La Palma is a very steep island!
Take a (light weight) jacket on your hikes, it gets cold in the mountains.
A rain poncho or rain jacket is also a good thing to take on your hikes.
sweater: nights can be cold, especially in the mountains
bikini: to relax on the black beaches
Toiletries and Medical Supplies: sunburn lotion and after sun lotion
Photo Equipment: as much as you can take, because almost everything you see is worth a photograph.
If you're going for a hike on the La Palma island, don't forget to take a warm jack with you. Up on the mountains, but also around El Paso, it can get pretty cold and windy, and you'll be glad to have a warm jack.
If you're doing the Volcano Route, wear shoes that close; because of all the loose sand.
Carry plenty of water with you! When the sun gets nasty, you'll be needing a lot.
Luggage and bags:
For hiking you don't need much as it isn't big, just if you do the round the island hike, you will need a bigger bag for hiking, otherwise a day bag is enough.
Clothing/Shoes/Weather Gear: Good hiking shoes are essential, as is a raincoat and a sweater.
Toiletries and Medical Supplies: It is a part of spain, so you can get everything here, so just the basics are enough to take
Photo Equipment: On the hikes, take a backpack with minimum off gear, and not a shoulder bag cause some hikes requiers you to climb high and too much gear is just a pain in the ass (I can know this, ouch)
Be prepared to any kind of weather, especially if you are in the mountains. Wear strong boots and always have some warm clothes with you.
Camping/Beach/Outdoor Gear: On the beaches you can find no shade and because they are black they get very hot, so that it can be wise to wear shoes. Bring an umbrella or anything else to have some shade.