Inverness is the nearest commercial airport. It's about a 2 hour drive from Skye. Glasgow and Edinburgh airports are alternatives but they are both, at best, a 4 hour drive from Skye. If you are lucky enough to have access to a private aircraft, there's an airstrip just outside Broadford!
I know of two options. Kyle of Lochalsh has a railway terminus which is reached by the line from Inverness. Mallaig is reached by rail from Glasgow and includes the famous West Highland line between Fort William and Mallaig, It is reputed to be the most scenic rail journey in Britain and featured in the Harry Potter films. That stretch includes the wonderfully picturesque Glenfinnan Viaduct. If you travel to Mallaig by train, you could consider using the car ferry to Armadale as a foot passenger and I can collect you from there by prior arrangement.
Citylink runs services between Inverness and Portree (Skye) and from Glasgow to Portree. You may find that this is your cheapest option for travel, especially if you have an over-60s travel card. However, it's also the slowest option - fine if you're patient or have several books to read.
I have spent many years driving to Scotland from the south of England so I feel I can offer some useful advice. A drive to Skye from London or anywhere along the south of England will take you at least 12 hours, even if road conditions are kind to you, and that's without taking time out to enjoy the scenery through the Scottish Highlands! It's far too much driving in one day for most people. I'd suggest the Lake District could make a pleasant stop-over area - or there's Gretna of course but I take no responisbility for any eloping!
You may be tempted to take the scenic route along Loch Lomond. I prefer this route myself though you need to be aware that the road towards the northern end of Loch Lomond (after you turn right at Tarbet) is narrow and twisty, and even has a single track section controlled by lights. It's not a road for the faint-hearted or large vehicles, though you will find vans and lorries using it. However, the scenery around Loch Lomond is very pleasant. After Luss there are public toilets at Firkin Point as you approach Tarbet.
Leave yourself time for a stop at Glencoe if that's on your route. This is a broody and atmospheric spot, whatever the weather, and especially if you know about the history attached to it.
If you've taken in Glencoe or you've come up through Oban then you will find yourself driving through Fort William a little later. If you are using satellite navigation, please make sure it doesn't take you left at a roundabout as you leave Fort William. That would take you to Mallaig and the only way to Skye from there is by ferry to Armadale. Stay on the A82 signposted for Inverness and Kyle of Lochalsh (A87). If you can, it is a good idea to switch off the option of ferry crossings on your sat nav to avoid this costly mistake. You should pass the golf club and the ski run on your right as you leave Fort William, though they can both be a little difficult to spot.
If you want to pick up provisions, Fort William is the last place you'll find with large supermarkets (including Morrisons and Lidl). Broadford has a Co-op which sells a reasonable range of food and household items. There's also a Co-op in Kyle of Lochalsh just before you cross the Skye Bridge.
I have given directions from the Skye Bridge to my house on my Contact page.