Hi John
I fish the westside lochs and burns (Tingwall - Sandness road) when I’m on Shetland with work.
Make sure you pick up a copy of the Fishing on Shetland leaflet from the Shetland Times bookshop when you arrive.
You can get an annual permit (£30) from LHD on the harbour front in Lerwick. DH Marine up by the marina, and Ocean Tackle in Brae, also have a decent selection (but pricey).
Standard tackle and leader set up work on the lochs. Limit your cast to two droppers/flies if it’s windy. And don’t forget to try a Foam Daddy from August onwards.
It’s worth having a cast in the burns/voes for an early seatrout, although I’ve had most luck in September for them. You might want to consider taking spinning rod for the days when it’s too windy for flies.
I haven’t fished up by Brae so can’t give you much specific advice, other than to say most of the lochs seem to hold fish, so if your struggling on one loch just walk over the hill to the next.
If I had to choose my favourite locations, it would be the Loch of Clousta for brownies, and Peerie Spiggie beach for sea trout.
Have fun - the locals are often happy to share information on good spots, and the Trout Fishing in Shetland Facebook page and SAA website are both excellent resources.