Only 2 days - so much to see such little time.......
Portree will be useful as central base. Suggest you should aim to see:
Dunvegan Castle on west - expect queues though if you are going shortly as popular coach tour destination - nice tea room though...
Talisker Distillery - souvenirs galore......
Museum of Island Life at Kilmuir is interesting, and there is also Flora MacDonalds grave to visit - start hummimg '...speed bonnie boat.....'
Armadale Castle has island life exhibits too
Get to see the Quirang as well on the Trotternish peninsular - if its a clear day the scenery is what Skye is all about.....
Eat at the Three Chimneys - miss this and you'll kick yourself. More awards than you can shake a brochette at........
And if you are driving to Skye - don't go over the bridge - boring design that adds nothing to the experience - go through Glen Elg and take the tiny ferry across - otter hide close by, seals, loads of other wildlife and a nice drive through wilderness scenery taking you to the main road to Portree.
If you break the journey to Skye, stay at one of the little B & B's in Glen Elg and eat at the Glen Elg Inn - good food and full of character. Locale is where Gavin Maxwell sourced ideas for Ring of Bright Water and the otter stories, and there are some ancient brochs and even Roman remains in the area to view as well, so plenty of things to do on the way.
Have a nice time...!