Just to summarise, and possibly add to, the above:
The Wightlink ferry from Portsmouth to Fishbourne leaves from the terminal immediately adjacent to Portsmouth Harbour station. The crossin takes about 45 minutes. However it's a 15-minute walk to the nearest bus stop, which is at Kite Hill and served by Southern Vectis route 9. Otherwise (unless someone is meeting you) you'll need to find a taxi. (Wightlink ticket office staff can suggest phone numbers).
The Wightlink seacat service also leaves from Portsmouth Harbour but operates to Ryde Pier Head, taking just 22 minutes. From there, Island Line trains run to Brading, Sandown and Shanklin or you can easily walk to Ryde Bus Station for connections to destinations across the Island.
Alternatively, instead of travelling to Portsmouth Harbour, you can take a train to Southampton Central. From there you can take a free shuttle bus (running every half hour) to Southampton ferry terminals. Red Funnel operate a conventional ferry from Terminal 1 to East Cowes (taking 55 to 60 mins) and a hi-speed ferry from Terminal 2 to West Cowes (taking 23 to 25 minutes). Buses connect with both services on the Isle of Wight.
The fastest crossing to the Isle of Wight is from Southsea to Ryde using Hovertravel's hovercraft service (10 minutes). To get to the terminal at Southsea you take the Hoverbus service from Portsmouth rail station. (£2 single, £3.50 return). Connections in Ryde are as above.
There's also a Wightlink ferry from Lymington to Yarmouth (40 minutes) but it's not easy to access by train from Chelmsford. (You need to get to Brockenhurst and then change to a service to Lymington Pier).
NB: Ryde Pier Head, Ryde Hoverport, East Cowes, West Cowes and Yarmouth (IoW) are all destinations recognised by the National Rail ticketing system. You can buy through tickets to any of them from Chelmsford, which will include the fare for the relevant crossing.
I've travelled from Suffolk to the Isle of Wight by train several times (which takes me through Chelmsford). I like to use the hovercraft from Southsea but that's possibly just because I'm still a big kid at heart. Otherwise I'd use the seacat from Portsmouth Harbour.