I live near Ipswich and I've travelled to/from Gatwick Airport many times.
By road, Mushroom25's route is the only sensible one.
By rail, I'd consider Mushroom25's route only if going in the other direction (i.e. from Ipswich to Gatwick). That's because mainline services from Ipswich to London usually call at Stratford, whereas northbound trains don't. (So you'd have to change again if travelling northward).
A much easier rail route is to take a First Capital Connect train from Gatwick to Farringdon. Then take an eastbound Underground train, for 3 stops, to Liverpool Street. (It's irrelevant as to which line the train is on. All three lines through Farringdon go to Liverpool Street). Lastly take a National Express East Anglia train direct to Ipswich.
There are no direct coach services from Gatwqick to Ipswich. To complete the journey by coach you'd need to take a (frequent) service to London's Victoria Coach station, followed by a (far less frequent) coach to Ipswich. If the timings din't work out it might be just as quick to get a coach from Gatwick to Stansted and then board the (2-hourly) service to Ipswich. However, unless you're really hard up, coach travel isn't really a practical option.
Chris