Tokyo Tower is a good place to visit in my opinion. It is one of the well-known symbols of Tokyo and retains an old-fashioned charm. The tower itself is actually a radio tower and was built in 1948. It stands 333 meters tall and is a distinct reddish orange color when not lit up.
One of the things to remember about Tokyo Tower is that it looks best when viewed from a bit of a distance. Although it can be fun to go inside and pay for a ticket to go up to the top of the tower, the view is comparable to views from other tall buildings in Tokyo. Much like Mt. Fuji, Tokyo Tower is best enjoyed from outside.
There is a temple called Zojoji temple in front of Tokyo Tower. It's interesting to see the temple and the tower together, and this place is a good place to take a few snaps for instagram. The temple grounds are open 24 hours, so it is also possible to get some night views of Tokyo Tower from here. You can also enter the temple until 5 pm. There are sometimes Buddhist ceremonies taking place and the interior is beautiful.
Below are some shots of Tokyo Tower lit up at night. The colors and style of the lights change according to special events and seasons. More details can be found at the official Tokyo Tower website. Skytree, the other famous tower in Tokyo, is about twice as tall as Tokyo Tower. However, Skytree is lit up with LED lighting, whereas Tokyo Tower has non-LED lighting which looks warmer and more romantic in my opinion.
There are two levels within the tower, and a standard priced ticket only gets you up to the main deck level. You have to pay extra to go all the way up to the top deck. If you don't want to spend the money, then my recommendation is to view the tower from below, without entering and paying for a ticket. By strolling around the tower you get great pictures, you get to experience the atmosphere of this well-known landmark, and best of all, it is all free.
Sorry, the picture below is quite old, but the tower itself is the same! :)