Times Square is one of those places where if you are visiting NYC for the first time (or maybe just in for a weekend) you want to check out. When you hear phrases like "New York never sleeps" it is because places like Time Square are filled with many people at any day and at any hour. Many of the stores are open very late and vendors are trying to sell you anything from custom art to portraits to pretzels and everything else under the sun.
When I visit I never leave empty handed and that is true of most of the tourists. One year I bought a custom portrait. Another time I bought a t shirt. The best travel advice is to bring your own water or drinks (the CVS in Times Square charges nearly 5 dollars for a bottle of water) and to watch your bags.
The most famous show is the ball drop on New Year’s Eve but if you are not a fan of that there is always a Broadway show just around the corner.