Below is a detailed list of the most interesting things to do in this immense square. It's also very popular to watch the daily raising and lowering of the flag by the honour guard (ask for the exact time at your hotel, as it changes throughout the year). The downside is that both, in the morning and in the afternoon, there is usually a large crowd, so it's a good idea to arrive well in advance in order to be able to see.
While many tourists go up into Tiananmen Tower (the original wall of the main entrance to the Forbidden City), we don't recommend it, as there is nothing really remarkable about it (at any rate, it costs 15CNY).