There usually isn't anything much special during the day. A few years ago they did a kids thing at noon in DCA, but that was only one year. At night starting about 8 or 9 they have music in the different areas. They usually have cameras from the Dick Clark Rockin' Eve near Small world. We were there last year. They will usually throw stuff out for the crowd for the cameras. Balloons, pompoms, signs stuff like that. My daughter danced with goofy for the longest time. We taped it and were able to see her a few times.
In addition to the regular fireworks they have fireworks at midnight. They have different bands in the different areas with countdown. Usually small world has radio disney/dick clark taping so you will see characters here, Tomorrowland castle, and rivers of america. They all have different themes. Last year they also had fireworks at DCA I think. They also give out hats and horns. They start around 7 or 8 if I remember correctly. They have lots of spots all around the park, and people line up when they first start, but after about half an hour or an hour, you can just walk right up and grab some. They have tons throughout the park. We generally get at least one extra horn as our daughter usually manages to destroy hers by midnight.
We go almost every year.
It gets extremely crowded, but even though they threaten they have not had to close the gates to guests. Lines are long. We usually don't ride much. Mostly we walk around and listen to the music and stuff. Lots of people arrive between 6 and 11 don't so expect long lines for parking, tram, and entrance. Most years they have stopped the trams and had people walk. Last year, they started the trams up afterward. A lot of stuff stops around midnight so that the CMs may celebrate (they will temporarily stop running a ride but you can stay in line)