I have visited Shetland twice. I would recommend May or June. I flew from Manchester both times via Aberdeen, using Flybe who offer a good service. I had no problem with midges. I hired a car on both occasions and stayed B&B then holiday cottage, both good.I think you need two weeks to get around and visit other islands.
The Shetland website is very helpful and provides a list of walks. Note that many walks are rarely visited and paths indistinct.
Walks I would recommend:
Bressay
Mousa go to see broch
St. Ninian's Isle
Sumburgh Head for seabirds (many puffins)
Esha Ness (coast)
Ness of Hillswick
Ronas Hill(park by radio towers on Collafirth hill)
Isle of Fethaland
Hamnavoe/Fugla Ness

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There are a few gardens which are remarkable for so far north. Lerwick park gardens even on the 1 October.
Places I would like to visit in the future:
Unst (you need to stay a few nights/it takes half a day to get there)
Foula use internal flights.
The highest temperature is about 18 centigrade.
If you want more details to contact me.

Thank you for the info a trip to start planning