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Your UK visa refusal should have little to no bearing on your attempt to get a US visa - it is unlikely US immigration officials will even know about it. Staying in the US varies greatly based on where you are staying and the level of comfort you desire. You could easily spend several hundred dollars a night on a hotel in downtown New York City. At the same time, there are cheap solutions if you look for them. I've found private rooms in a hostel in NYC before for about $60/night.
Upvote:2
You can apply for a B-2 visitor visa. If you're worried about the visa process, this link offers a step-by-step guide for obtaining a visitor visa to the USA: http://www.path2usa.com/visitor-visa-guide, and walks you through each step of the process so you're fully prepared. You shouldn't have a problem getting your B-2 visa as long as you know properly complete the required forms and know what to expect during your interview. As for the costs of your trip, you can find hotels, hostels, or airbnb.com rooms for pretty inexpensive (some as low as $50 per night) depending on the city.