How to get from Picadilly tube station to Southhampton cruise port?

Likely the Underground to Waterloo, then train (Southwestern Railway) to Southampton. OR National Express coach service from Victoria Coach Station.

Southwestern Railway is the train operating company covering that route -- https://www.southwesternrailway.com/

Travel from London Waterloo (WAT) to Southampton Centeral (SOU). There are a couple direct trains per hour with a 1:22 journey time. Advance single tickets that tie you to a specific train departure/journey can be had as low as £20 to £26, while an Anytime ticket is £54.50.

To get to London Waterloo station take the Bakerloo Line southbound from Piccadilly Circus to London Waterloo.

There will always be a dichotomy between best and cheapest.

Best will always be a door to door driver service.

If you book a month or so ahead of time on a specific train service from Waterloo to Southampton Central is £14 Advance Fare 2nd class on morning trains (less in the afternoon) or £19.80 1st. 1st class gives you no perks other than a quieter part of the train, so a bit more room. Plus tube, taxi or bus fare from Pic Circus to Waterloo.

National Express Bus from Victoria turn up on the day is £11.50, book a month or so ahead is as low as £5.20. And the driver looks after your luggage. So is that better than the train?- maybe. But of course you can't walk round as you would on the train. 2:20 journey at 0830 or 1030. Take the 0830 in case of issues on the motorway/freeway.

At the Southampton End very many Brits (including me) would not even think of taking a taxi, as the main cruise terminals are so close to both the bus and rail stations. Even with luggage it is very walkable. Taxis are not hard wired into me.

But most people on here use a taxi, and there are a number of complaints on here about the price of the taxi. The drivers have a captive market.

So book ahead, use public transport to Victoria then walk at the Southampton end, it is under £10.

Likewise, but on the train it is under £20.

Door to door with a driver will be no faster than the train due to the long slow creep out of London in the traffic, and will be somewhere into a three figure cost.

Just depends on your priorities.