You might want to have a look at Walk Highlands website to get some ideas for some good walks. The places you have mentioned on Skye are what I call the 'honey pot' sites that everyone who comes to Skye visits, mainly because of Instagram and lazy journalism that concentrates on the few well known spots at the expense of other equally beautiful places on the island. You will find hundreds of walks described on the Walk Highlands site that will hopefully help you to find your own special places on Skye.
If you are set on going to Neist Point for example, be aware that the car park will be very busy in the couple of hours before sunset.
The Fairy Glen is always busy, there is now pay and display parking and you must not park in the passing places on the single track road leading to the glen. Likewise parking at the Quiraing also now requires payment. A much less busy walk but with equally fantastic views in the same area can be done by parking for the Quiraing but heading up the hill on the other side of the road.
As you are staying in Portree, this fantastic hill walk is a short drive from Portree. Ben Tianavaig Walk.
One thing that you definitely won't want on Skye is an umbrella. Nobody here uses them as it is invariably too windy. We all just wear waterproof jackets and hats. If hiking you will want to carry gloves and waterproof over-trousers in your backpack.
For driving, bear in mind that there are lots of 'single track roads with passing places' on Skye. You need to know how to drive those roads. Basically there is only enough road for one line of traffic. If you meet a vehicle coming the other way either you or they have to give way, using the passing places to get past each other. If the passing place is on your left you drive into it to allow the other vehicle to pass. If the passing place is on your right you stop opposite the passing place, the vehicle coming the other way pulls into the passing place and you then proceed. This might help. Video of how to drive single track roads.
Hope that helps! Feel free to fire any more Skye related questions my way.
Jacqui (Skyegirl)