My extended family have been on all the major cruise lines and overall you can't beat P&O for value for money, family friendly, choice of ships, and choice of destinations from Southampton.
Going from Southampton is so blissfully easy, you can take as much luggage as you like.
For 7 people you'll need 2 cabins. If you can afford it 2 adjacent balcony cabins are great because the balcony divider can be opened so there are in effect connecting cabins without being official connecting cabins with an inside door (not many official connecting cabins).
A cheaper option is a balcony cabin with an inside cabin across the corridor.
Personally I think a balcony is worth it because just looking out to sea is amazing, others say it's only for sleeping, lots of deck space so don't bother.
I recommend talking to a specialist cruise agent, we use Bolsover who are brilliant, they can check exactly what cabins are available and also shave a bit off the official price. They also answer the phone quickly and are great if you've got a query. I've heard of many frustrations trying to book direct.
To get adjacent cabins you'll need to do 'select' fare not saver as with a saver you could end up with 2 cabins at opposite ends of the ship. However with select you'll get various perks and on board spending money which often makes it barely more expensive.
I'd go for Iona or Arvia as they are new and sparkling. The ships themselves are so amazing with the wow factor.
Whatever you do you'll have an amazing time.