Have a great trip. For travel from Heathrow to a hotel in London, it will depend on how much baggage you have and whether you are happy to use public transport. There are 3 easy options.
The Tube will take you all through central London. It starts at the airport, but can get crowded and hot.
The Heathrow Express Train takes you to Paddington Station.
A new option which just opened is the ‘Elizabeth Line’ which is a train into the centre of London. For a public transport route to any hotel or any other location, you can use TfL.gov.uk. One tip for anyone with luggage, look for stations with a wheelchair symbol, Those have step free access train to street - handy with bags.
Taxis are available but very very expensive. Uber is also an option.
To get to Southampton, I would take the train. There are at least 2 per hour. Bear in mind that despite being a busy route and a busy cruise port, the trains have no separate luggage racks (crazy decision). The journey time takes usuall 70-120 mins max. If you book in advance it is very cheap. 1st class is not worth the premium, you get nothing extra for it. Southampton Central is a short taxi ride from the cruise terminals. Alternatively, the National Express Coach service can also take you to Southampton from London.