Hi, jdcarlton79,
Another vote from me for Skye, Lewis, and Harris. Unless you are willing to fly from he West Coast islands to Orkney and Shetland, as jjgurley did, you'd be best served staying to one side of the country.
You'd need at least three nights on Skye. Take the ferry from Uig to Tarbert, on Harris. South Harris has some of the most beautiful beaches in the world. You could spend several days on Harris/Lewis, and not feel as though you've seen it all.
If you can include this in your travels, I would very highly recommend a trip out to St Kilda, which is about 50 miles west of Harris. There are two companies which will transport you out to St Kilda - Kilda Cruises and Sea Harris. Kilda Cruises runs more frequently. It's an all day adventure - three hours out on the boat; five to six hours on land; and three hours back to Leverburgh. It's better to leave from Leverburgh, as opposed to Uig, as it gives you more time on land.
You will definitely see puffins. There are thousands of them on the sea cliffs. On the return journey, the boat will take you around Boreray and Stac an Armin. It's a truly amazing place, and a UNESCO World Heritage site. It was on my bucket list for years before I finally got there.
If time doesn't permit a full day on St Kilda, you could consider taking the ferry from Leverburgh to Berneray, visiting North Uist (and possibly Benbecula and South Uist), then heading back to Skye via the ferry from Lochmaddy to Uig. There are several good hikes on North Uist (and many on Harris, as well).
As far as the midges go, we've had pretty good luck with Repel, which is available on Amazon, and is also carried by Target. June is definitely midge season ( It starts in May.), but you can get midge hats in most outdoor stores. They run about 12 pounds. We bought the Lifesystems brand a few years ago, and they have proved a godsend! I have to say that you need to have an Engineering degree to get them back in to their carry pouch!
I hope that you have a wonderful time in Scotland. You need to solidify your plans for the Highlands and Islands fairly soon, as lodging is not that plentiful, and books up fast!
Very best wishes,
Mike (Auchterless)