DW and I do an annual trip to DC. Others have already posted that you need tickets only to a couple of things that might be of interest to you. Otherwise, the monuments and museums are free.
Definitely take the metro everywhere. Just a note --- if you go to the DC metro website you'll see that there are weekly, unlimited ride tickets you can purchase. DW and I like to go to multiple areas in a day rather than spending all day in one area. Sometimes, we'll hop on metro to go to a specific restaurant.
So, we'll hop on/off the metro to different locations. If that is for you, you'll want the unlimited ticket.
I have no idea how you travel or what you like to do. We split our time every day, weather permitting. We'll often do a long walk for half the day, and other half go to a museum. For instance, walk around Georgetown early (before all the crowds hit) in the morning, and back the the Mall in pm. Or, Old Alexandria, again, in the morning, while it's fairly quiet.
Anyway, have fun planning and visiting...........