With difficulty. You will have to take the overhead train from Colchester to Liverpool Street Station, then the underground to Victoria Station and then the overhead to Gatwick.
Gatwick Rail Station is at the South Terminal but there is a small shuttle train which runs between the north and south terminals.
Assuming you meant public transport: You can get a Thameslink train from Gatwick Airport Train Station to St. Pancras station. Then a short walk to Euston station where you can catch a Virgin Trains service to Stafford (the main station in Staffordshire).
About 35 miles and there is a very good train service which runs between Gatwick Airport and Victoria Station in London.
The train from Gatwick Airport to East Croydon Station would take about 20 minutes.
There is a fast train service from Victoria Station.
You can't travel directly from Gatwick to Paris by train. You will have to take a train to London St Pancras and then get on another train to Paris. Unbelievably, you can't fly from Gatwick Airport to Paris either.
Train from Colchester North Station to London Liverpool Street
Gatwick Express. The train goes to Victoria Station and the journey takes about 30 minutes.
Take the train from Cardiff Central to London Paddington, take the Circle Line to Victoria station, and catch the Gatwick Express to Gatwick Airport.
You will have to change trains in London.
By train, take any northbound train from the station (don't take from the Gatwick Express, or any train destined for Reading) and get off at East Croydon.
Eastbourne doesn't have an airport - the nearest is Gatwick, so you would have to fly from Edinburgh to Gatwick and take a train from Gatwick to Eastbourne.