The airlines which offer direct flights London - Cagliari Elmas are: EasyJet (Stansted) and British Airways (Gatwick), both with only one flight time a day.
For another option, often a bit cheaper, but with one stop in either Rome or Milan, you could also fly with Alitalia.
If you are interested in considering the rental of an apartment or a villa you can look on airbnb, homeaway, holidaylettings and subito.it.
For hotels and B&B's, below are a few options:
Taxis
Taxis can be found outside Cagliari aiport and will take you to the centre in 15-20mins for around €15-18.
For those looking to explore Cagliari and its beautiful beaches it would be advisable to rent a car.
These are some companies you can find at the airport:
Those staying near to Poetto beach can easily reach the city centre by bus in 15 minutes.