I went to Shetland 2 years ago at the beginning of May. I only had a week and, because I wanted to cover as much ground as possible, that ruled out taking the bike as I wouldn't have had time to explore off to the side and sit soaking up the view and wildlife. I got the train to Aberdeen, ferry to Lerwick, and picked up a hire car at the ferry terminal. I took my one man tent and camped at camp sites.
Gradients - The hills on the main roads aren't too steep but tend to be long and draggy, so don't think you will have any issues with your gearing. The wind will be a major factor though. Minor roads have steeper bits.
Midges - No issues at that time of the year.
Camping - I stayed at 4 different sites listed on the Shetlang.Org website
here. All of the sites were clean and well maintained, and if you are discreet have somewhere you can cook your meals indoors if the weather is iffy.
- Bridge End - I was the only person there. Unmanned. Sheltered.
- Braewick - A bit exposed as is on the top of a cliff, so you need a good tent and double-pegging, but worth it for the stunning views. Walked down to the beach in the evening and watched a pod of 3x killer whales cruise past 50 yards offshore. Great showers. The wooden wigwams attract some strange but harmless people. That whole area of Shetland, Eshaness, is stunning.
- Gardiesfauld - Brilliant. In the grounds of a youth hostel so you have access to kitchens/showers etc. Sheltered.
- Burravoe - Really good. Bit of a steep hill down to it. Unmanned. Annoying twat in a camper van who was an authority on everything.
Shops are few and far between so plan your food purchases well.
The fish and chip shop in Brae is brilliant.
Jarlshof is overpriced for what it is (imho).
The sea cliffs at Hermaness with their Fulmar colonies are an assault on the senses.