Visiting the States versus living here can be very different, just as many Brits tell Americans who ask questions about moving to the UK. I suggest you do some research before you try to move to the States. You would have options, but most are expensive unless you happen to get a good scholarship. I think the most common way for younger people from the UK to get here is by studying here - I had an English student when I taught at a university. It was also the way I, as an American student years ago, was able to live in France for a while - I had a good scholarship and I studied there for a few months. That may be the better initial course of action for you. I am honestly not sure how easy it would be for you to get a visa or green card, but I can say that as someone from the UK, you should have an easier time being admitted here than people from other parts of the world would. But again, it would be a good idea for you to either come here as a student or have worked a few years and gained some experience. There are so many double-income families here, so in my opinion, the demand for nannies here is good. I'm not trying to be sarcastic, but I'm sure some upper middle class or rich couple would love to hire you as a nanny mostly because of your accent; they'd probably think you sound more sophisticated, to be honest. There are also lots of parents I know who work outside the home who prefer to have in-home care for their children, so that's a plus for you.
But again, don't have your head in the clouds - be realistic about all of this. It's hard to know how much it will cost. The good news is, if you go to a university here, you have a myriad options, with many good schools that are still affordable. When I moved to France as a student, I basically only took my clothes and a few books I figured I'd need. I bought everything else in France and stayed with a French couple who charged cheaper rent. There would be plenty of rental options for you (for instance, I see lots of people in the classifieds of my local newspaper offering a single room to rent, etc.). For buying furniture, you can save yourself a boatload of money and go to things like estate sales, which people in suburban neighborhoods are always having, where you can get furniture and the like there for dirt cheap and still in good shape. It really depends on how long you'd stay and what type of job you'd have though. Finally, pay attention to the area of the country you'd like to live. Be open minded, and don't just consider places along the east coast or west coast, all which tend to be way too overpriced. I was born and raised in a Midwestern state, so I'm always promoting this area of the country. :) It is very reasonable cost of living in the midwest too. You would be welcome in any part of the country though, I guarantee it. We tend to be quite fond of British people here. You will experience some culture shock, to be sure, but having been in parts of Britain myself (and loved it!), I found that it was the easiest place for me to adapt to, much easier than France and Italy at least. I think many people who have not been to both the U.S. and Britain would be surprised at how much there is in common.
Finally, while I gave a thumbs up to one other post here, I don't suggest you rely on an American man to marry you. ;) Well, if you find a good one who truly loves you, go for it, but otherwise, have fun and let this experience belong to you. Personally, I think men in the UK are much more polite in general...but that's not to say there's no good American men, because there certainly are some really nice ones. I managed to get a good one! :D
Hope this helps, and good luck to you!