Plan your living arrangements first. There are amazing hotels in New York City, and most of them are well known. Popular ones right now include the London Hotel, the St. Regis, W Hotels, Starwood, The Plaza, etc. If these are too expensive for you, there are always more chains that are a step down price and luxury-wise. However, my suggestion would be to check out Craigslist for a short-stay local loft. This is a great idea because you can get an amazing place for much lower than the cost of a hotel would be, plus you get the tips of a local in terms of neighborhood places to eat and go out. In addition, you have more privacy than you ever would in any hotel.
Make a list of what you want to see tourist-wise. Think about Ellis Island and the Statue of Liberty, the Empire State Building, Rockefeller Plaza, Radio City Music Hall, etc. You may not have time to see everything, but I wouldn't recommend a see-all tour bus. Remember, this is supposed to be romantic. Choose one place "with a view" because these are always fun to share with someone you love. For the statue of liberty, do cruise from the South Street Sea Port. I went on a small sailing cruise that was the romantic thing my boyfriend and I had ever done before. We saw beautiful views of the city and the Statue of Liberty, sipped wine, ate cheese, and never had to stand in a line. It couldn't have been more perfect.
Go to a park. Central park is always popular (pictured). My personal favorite is Bryant Park. The wich'craft stands are amazing, it's small, quaint, and beautifully shaded. Pack a picnic and sunbathe in the park; nothing is more romantic to do during the day than this.
Visit a museum. Guggenheim...the Met...MoMA...American Museum of Natural History. They are all great, all historic. For something romantic, I'd go with the Rubin Museum of Art. It is the most beautiful museum in the city (the re-done Barney's building) and it has a vast collection of Himalayan art. Every friday the museum hosts their K2 lounge, which offers drinks, a movie and admission to the museum all with the purchase of a ticket! Art couldn't be more hip, fun, and trendy than this.
Pick your restaurants. There are so many great restaurants in NYC that it can actually be overwhelming. Here are some great ideas for date places, organized by cuisine: Asian/sushi: Thai Angel (on Grand St $), BONDST on (Bond St $$$), Japonais (on 18th $$$) Italian: Dell'Anima (on Jane St $$), Da Silvano (on 6th Ave $$$) Vegetarian/Vegan: Teany (on Rivington, $), Candle 79 (on 79th $$), Quantum Leap (throughout the city $ and ****) Also check out New York Magazine (www.nymag.com) for a vast array of reviews.