Hi Kathy, yes, there are regular trains from London Waterloo direct to Southampton in about 90 minutes, or slower stopping trains taking up to 2 hours.
From the Central station all of the docks are within a 10 minute taxi ride, and depending on who you are sailing with, there may even be a courtesy transfer. The only downside is that obviously the taxi drivers are there to make a living and certainly will try it on with cost, so you may want to ask before getting in rather than feeling ripped off when you unload. On the bright side, for short trips and they do overcharge, then you can be "very British" and not give a tip. They don't expect them on short trips, so won't say anything, other than "have a lovely cruise".
On the return (sorry to think about that) there are taxi ranks outside each of the terminals, but if you want to organise a return taxi to the airport, you may want to do that in advance - it should be around £80.
Please note there are 4 cruise terminals in Southampton, so make sure you know which one you want.
All the best,
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