Here is a list of my absolute favourite spots to visit on the Isle of Skye. I saw all of this with a day and a half of the island. While two-ish days is a reasonable amount of time to get a taste of the area, there is so much more to see and do that you could easily spend a week exploring.
Having hiked in quite a few places in the UK, I can confidently claim that Skye is one of the most magical! Without further ado here are my recs.
The Fairy Pools
The Fairy Pools are a series of waterfalls and pools flowing out of an impressive mountain range. The path follows the weaving river from the valley to the base of the mountains. The surroundings are gorgeous and the pools are so fun! You can jump in anywhere you want (if you are brave enough). The first big falls (pictured above) have a fantastic cliff jumping spot to the upper right of the photo. We saw some kids in wetsuits that were surfing the rapids too!
The Old Man of Storr
This is probably the most well-known site in Skye and for good reason! The "old man" is the prominent rocky pinnacle that sticks out of the hill (Storr). We went up just before sunset on a clear day and got to see the island bathed in golden light. There is a fork at the beginning of the trail and I recommend sticking to the right! There are also small trails that branch off of the main path as you get up into the pinnacles. I recommend exploring these as well as they mostly loop back to the main path.
Here are some more views and the little sheep friends that live in the area.
The Quiraing
This was the best thing we did on Skye and one of my favourite hikes in the UK. It is a 7km loop of a section of the Trotternish Ridge. You walk through pinnacles, up cliffs, and over bogs on the ridge itself. It is so fun (and very squelchy on a wet day) and has amazing views. I felt very epic while walking it like I was part of the Fellowship of the Ring embarking on a long adventure. While on top, I understood why the Isle is called Skye. The ocean was reflecting the color of the sky exactly and it looked like we were on a floating island resting on clouds.
Neist Point
This is the westernmost point of Skye and takes a while to drive to from the main towns of Portree and Staffin, but the drive is beautiful and really fun (for those of us who think twisty one-lane roads are thrilling).
Parking was rough (don't park in reserved spots because you assume no one will use them -- they will), but the walk out to the point and the views along the coast were well worth the hassle.