Wannabestressfree - we're going on P&O. We chose it for the free night nursery (plus baby milk/baby food, steriliser, cot, bath - literally everything you need with a little one!) and Southampton departure. I can't think of an easier holiday with baby.
If you're considering a first cruise, my advice would be to choose your line and ship carefully.
Royal Caribbean & Norwegian offer a wide range of entertainment for the whole family (plus Carnival which don't sail in Europe). Disney is great for little ones, but comparatively expensive. P&O are excellent for infants (and great for kids too). If you want smart, chose P&O or Cunard. Most other lines are casual now and have opt-in formal nights. I think Celebrity is a bit more sophisticated than other lines and their food is a notch above plus they have some great bars. Most lines now have a number of extra charge speciality restaurants which can offer good value for the quality of food you receive.
Costa and MSC have a more European mix of passengers and excellent spa facilities on some really beautiful ships. Most large ships have numerous pools and free hot tubs, but the best ones are on Royal Caribbean, Celebrity Millennium class and some MSC ships (infinity pools, large heated thassotherapy pools).
The quieter ships tend to be the smaller ones and these tend to attract an older crowd, such as Azamara, P&O adult only ships, Fred Olsen and HAL (the latter is favoured by those who like a traditional wooden promenade). They tend to be more sedate.
Tips are indeed extra and you need to factor this into price (don't be an arse and remove them as the staff rely on them) e.g. P&O £5.50 pppd. Alcoholic drinks and sodas are extra but they are usually equivalent to UK bar prices. Some lines allow you to bring alcohol to drink in your cabin (P&O, Disney). A couple of lines (Pulmantur, CDF) offer all inclusive alcohol but they are Spanish/ French speaking and older ships. Also many lines (Norwegian, Celebrity, RCL) frequently offer free drinks packages and the Italian lines sell very cheap drinks packages.
Then there's the luxury cruise lines, which are out of most people's price range (Regent, seabourn, windstar) and adventure vessels to Antarctica.... if you can afford these, then lucky you!!
A good place to look for info is cruise critic. For updates on cruise fares check out cruise fish net. I would not book a cruise with a high street travel agent - you will pay brochure prices. Online travel agents (or direct with line) will offer best prices and often chuck in on board credit, but be aware they may not be fully bonded (US cruise agents such as Vacations to Go aren't bonded - which may or may not be an issue for you).