I think Sumburgh Airport to Lerwick is £2.90 single.
travel.shetland.org
The airport bus goes past Jarlshof which is a fairly famous ancient village. If there’s no-one in the ticket booth, you can still get in anyway. It’s all open.
https://www.historicenvironment.sco...rehistoric-and-norse-settlement/prices-times/
There’s a museum in Lerwick. You can take a bus out from Lerwick Bus station to the Northern Isles. A good cafe/restaurant/wine bar is Fjara on the seafront in Lerwick. (It featured in the closing episode of the most recent series of Shetland where Tosh & the DI go for a glass of wine and decide to have a bottle instead). The food’s very good there and you get a magnificent view across Lerwick harbour to Bressay. Seals and the odd otter are often seen from the cafe window..
https://www.fjaracoffee.com/
The most recent series featured quite a lot in Brae (25 miles north of Lerwick). It has a very highly rated fish & chip shop! Busta House hotel (the former Lord Lieutenant's home) is a good pub/hotel. If the bus is going on from Brae to St Magnus Bay, ask the driver to drop you at the end of the road to Busta.
Be aware that not every tourist spot is open in February. And the birds that many folk come for are generally seabirds that only come ashore to nest April - August. St Ninian’s Isle is famous. It’s a tombolo (a 2 sided beach) and has been in many episodes. It’s probably accessible by public transport. It’s near Bigton. I think there’s a minibus that connects with the airport bus several times a day. Service 7 Bigton & Quendale feeder.
https://www.visitscotland.com/info/see-do/st-ninians-isle-p2571251
You’ll get some scenic views from the service buses that go to places like Walls, Brae and so on. Wrap up well and be prepared for rain. It’s generally wet and windy in February. And daylight is very short.
Be aware that not all of Shetland is filmed on Shetland. A lot of the scenes are done on the mainland, often around Glasgow! Any outside shots showing mature deciduous trees in the background are highly suspect. But the overall scenery is generally genuine enough. D/I Perez’s old home on the waterfront in Lerwick is easy enough to find. There’s a little short beach beside it. Don’t knock on the door as the real residents have had more than enough visitors!
The overnight ferry is very comfortable but it can get lumpy in February. The vessel could get to Lerwick from Aberdeen in about 7 or 8 hours but that would mean arriving at 03.00 which suits no-one, so what they do is switch one engine off for half the journey and cruise at about 10 knots, so they arrive around 0600. It can be good night’s sleep but a longish journey.