Take a trip to Il Laboratorio del Gelato on the lower east side (on Orchard Street between delancy and broome) for the best f*cking gelato in all of Manhattan. They always have interesting flavors which change daily (my favorite flavor is rice). A perfect hot summer day treat, $3.25 for a small (one or two flavors). You could also stop in at the Lower East Side Tenament Museum, right across the street.

Central Park is great (and free). Many museums have "pay what you will" days. There are tons of cheap things to do in New York. You might want to pick up a copy of Time Out New York (a weekly magazine) when you get here, which has hundreds of listings of things to do around the city that week for every taste and every price range (from free to outrageously expensive)