We live here and often use the Metro to get around. The buses are part of the same system, but I'm less familiar with them. When you arrive, go to the nearest Metro station and buy SmartCards for each person. You can use these to pay for any Metro or bus ride, so it's nice to have them in hand and loaded with funds.
There is a purchase fee for the card and then you can choose how much money to load onto each card. I would maybe do $10 per card to start, since you won't be here very long. You use it when you get on the metro (or bus) and then the fare is subtracted when you scan it as you exit. It will tell you how much money is left on your card each time you scan it, so keep an eye on it and add money as needed. Depending on the distances, each ride might be $1-$3.
There is an app for the DC Metro & Bus system, and there's also lots of information on the website. I use Google Maps on my phone (or computer) to figure out if I should walk or use the Metro to get places. You can change the settings to show public transit as well as the usual stuff. When you ask for step-by-step directions from point A to point B, you can toggle between walking, driving, or public transit. I'm sure there are other ways to navigate. This is just what works for me.
Sometimes there isn't a metro stop near the place you want to go. We use Uber or taxis quite often too. It's often cheaper for 3 of us to pay one Uber fare (not during rush hour!) instead of 3 separate metro fares, and we get there faster.
For example, the Lincoln & MLK Memorials are about a mile from a metro stop, so we will use Uber to get there early in the morning on a weekend. Then we walk around the tidal basin while it's cool, and walk up to the National Mall from there. The Smithsonian museums usually don't open until 10, so we do one of those once they're open and then go home using the Metro stop nearest whichever museum we visited.
I hope this helps with the transit part of your question. I might be able to advise a bit on monuments, museums, etc. What are you wanting to see? You could stay a whole month and not see everything!