Transport from LHR to Southampton....

The National Express bus direct from all terminals is the simplest way to do the journey- https://timetables.nationalexpress.com/routes/NX/203/Heathrow_Airport-Southsea/O/pdf

I second using National Express. I've done this twice and it works great. For Terminals 2,3 you go to the Central Coach station. Just follow the signs. For Terminal 5 international arrivals it can't be easier. Exit out of International Arrivals, cross the terminal to the exits, cross a road, and you are at the bus area. National Express is on the far left as you leave the terminal, I think it is stands 11-15. (We were there in less than 4 minutes from exiting arrivals) You can book a ticket with "flex" fare which will allow you to change times if your flight is late. The beauty of National Express is the minimum of luggage handling. The driver puts your bags under the bus and pulls them out for you at the destination. No struggling up and down stairs in underground stations or train stations. When you arrive at Southampton, taxis are available. Our hotel (Premier Inn West Quay) was a five minute level walk away so we just walked it.

Other ways (more expensive) to get to Southampton:
Underground or Train to London, transfer to train to Southampton
Cruise Ship transfers. This requires arriving on the day of disembarkation which is a risky business unless you choose to arrive a day earlier and go back to the airport for the transfer.
Private Drivers: Most expensive, but for those that lack confidence for other modes, it will work.