You haven't said whether you'll arrive on a weekday or weekend. If weekday, don't be in to much of a rush to disembark. Have a leisurely breakfast. The taxi queue at the dock is marshalled and was very swift when I used it recently. Taxi to the station about £8. Early trains to London on a weekday will be very busy, trying to board with luggage won't make you popular. If you time it so you don't leave Southampton station until after 0930, you'll have much less crush on the train and benefit from (much cheaper) "off peak" ticket prices.
If you land at a weekend, any train is fine (off peak all the time) but engineering work can disrupt weekend services. That can be checked from 10/12 weeks before travel date.