The only formality is really the dinners, which are optional. Two nights are formal which is the ball gown/tux type thing, two or three informal, which is a cocktail dress/business suit, and one or two dressy casual nights. However, you don't need to eat in the dining room at all. You can have room service, use the buffets, or pay a charge to eat at Todd English (which I think is just suit and tie dress code). You can eat breakfast and lunch in either the dining room (where I think you just can't wear shorts or jeans). There's a pub lunch offered (not great at all but the pub is a great place to hang out) and afternoon tea.
It's not really all that formal. We spent most of our time in the pub or library. The ship is absolutely huge, so there is always places to go to get away, even when the weather is bad and they don't let you on deck.
If you're looking for the historical aspect, when they pull out of New York Harbour and pass the Statue of Liberty, it's a really intense feeling of actually doing the opposite trip your ancestors did, particularly if they are from the UK (which I assume from your nick they are). If you do choose to do the crossing, definitely take the time in NYC to visit Ellis Island.
If you don't have animals, there are other options for sea crossings such as going on cargo ships. A lot of people love these crossings, but they can take a long time. But it seems to be a much more rustic, yet intimate way to go by sea. For instance, people who are crossing all eat at the captain's table as there usually only a handful. It is up to you to entertain yourself for the crossing for the most part, but it does seem to be a bit more off the beaten path type of crossing.