Bring water with you. You will get thirsty.
Definitely dress in layers. Despite visiting off season when the weather was cold, I did get hot (and sweaty) climbing the stairs only to get cold when I was exploring each level. It is very windy as you get higher up the tower.
I found climbing to the first floor pretty easy but climbing to the second floor was a bit more challenging. I know this is common sense but pace yourself and stop at the platforms when you need to rest a bit.
You can take the elevator down because no one is checking tickets to go down but once again there will be a long line to wait. I found that I enjoyed the view of the scenery and the tower much better going up than going down – mostly because I was watching where I was stepping so that I didn’t go tumbling down the tower.