Passenger flights to Orkney are all operated by the airline Loganair
https://www.loganair.co.uk, based in Glasgow. There are flights from/to Inverness, Aberdeen, Glasgow and Edinburgh on mainland Scotland, and from 22 June 2018 Manchester. Loganair also flys to Sumburgh in Shetland and operates the inter-island air service to some of the smaller Orkney Islands. As part of the inter-island service, there is the world record shortest scheduled air flight between the Orkney islands of Westray and Papa Westray. If you want to experience this two minute flight, in a Britten Norman Islander, you will have to book well in advance though, as it is very popular. For getting about when you land in Orkney there are several car hire firms and one has a desk at Kirkwall Airport
http://www.orkneycarrental.co.uk. There is a regular No.4 bus service, operated by Stagecoach
https://www.stagecoachbus.com
to nearby Kirkwall, as well as a taxi rank.