Moving day will come sooner than you think, so get planning now.
It may or may not be practical to move out on the closing date. You may have to give the the keys to the new owner on closing day (Unless other arrangements have been agreed upon). So you may want to try and schedule the actual move for a day or so before closing. If you intend to move at the end of the month, contact a moving company or a truck rental company now before they're all booked. If you can move mid-week or mid-month, a moving company might cut you a deal. Go with a reputable moving company. Go with an established, insured mover, so your items are protected.
Pack it yourself, and pack early. Nobody will take the same care you will. Start early and spread it out over many days. Label all your boxes by room so the movers know where to put them, and label anything that's fragile.
While Packing ask yourself - do you really need to take that with you? Moving out is a chance to liberate yourself from stuff you simply don't need. If you haven't used it or worn it in the last year, you probably don't need it. Have a garage sale, or give it to charity.
Good luck and remember moving is a very stressful thing, so base yourself and plan your work.