I've done 2 cruises and enjoyed them both (one Mediterranean and one Norwegian Fjords) They were both through P&O.
As you have kids then I've heard Royal Caribbean are better as they have more stuff for kids on board etc. I've not been on one of their cruises but it's next on my holiday list to try as I've heard nothing but good things about them.
If you can get one that does a round trip from/to Southampton then you haven't got to worry about luggage weights for flights (or the added flight costs) etc which is one less thing to have to think about.
See if you can get free parking at Southampton thrown in and/or onboard spend so any alcoholic drinks can be bought with that instead. Depending on how much spend or how much you drink, could mean you don't really have to spend much more money when on board as food/none alcoholic drinks/entertainment is covered 24/7 within the initial cost.
Don't forget to budget for excursions if you want a guided tour of the places you stop at. This can vary on what your ports of call end up being. Some are the ports for the area you will actually be seeing, so you can get off the ship and do your own thing without paying for an excursion, but some ports of call are not the actual place you're meant to be visiting. For example, if one of your stops is Rome, then the port you actually dock at is Civitavecchia, which means a couple of hour road trip to get to Rome, so without actually booking one of the excursions, you could potentially miss out on seeing Rome.