The building off Farnborough Road can be identified by the 1950s Hawker Hunter two-seater aeroplane outside which can be seen from the road.
Trustee Richard Gardner said: "It is a huge, huge task because we've got about five years' worth of archives.
"And of course at the same time we've been preparing for the airshow exhibition because we've got two stands there this year, so it has been a bit of a nightmare."
FAST had hoped to be settled into its new home in time for the airshow but delays in getting the building prepared mean it will not be ready for visitors until later this year.
An opening ceremony is planned for the autumn but aviation enthusiasts will be able to see some of FAST's treasures before the autumn, at the Farnborough Airshow.
The major exhibition, on stand F1 in Hall One, will have an educational and heritage theme with important pieces from the collection.
These will include a section from the fuselage of a De Havilland Comet 4 and the "Beryl" compressor used in the Saro SR.A1 jet flying boat.
There will also be a smaller stand within the Aerospace Adventure Pavilion on the public days and three airframes loaned by QinetiQ - a Westland Puma, a two seat Harrier T4 and a "Raspberry Ripple' Jaguar B - on display outside in the Historic Aircraft Park.
Members want to raise funds for the trust as well as increase FAST's membership, which already stands at 800.
To find out more about FAST, including details on how to become a member, visit the website at www.fasta.co.uk or e-mail email@example.com.