Traffic jam in Beijing is notorious so I highly recommend using the subway system there to get around. The system is extensive enough that you should be able to get to anywhere in the urban area with it; certainly all the major landmarks and tourist attractions
It's also ridiculously cheap. Most of my rides cost me less than ¥4 (60 cents US), you can cross the entire city for ¥5-¥10 (70 cents-$1.5)
Using the system is fairly straight forward:
Have Chinese cash on hand to buy single tickets or the reloadable card. Make sure to have ¥5 and ¥10 notes with you as that's what the machines accept
I highly recommend getting the reloadable card, called Yikatong (一卡通). This card lets you simply tap into the subway system instead of buying tickets every time. You can get this card at any of the subway stations. The card doesn't cost money but requires a ¥20 deposit which you get back when you return it.
Use Google Maps to find your closest subway station and the one closest to where you want to go. Then use either Google Maps or this website/app to get the station-by-station transfers needed
At the station, just use one of the automated ticketing machines (it's got an English menu) to get your tickets if you don't already have Yikatong. Again, the machines accept only ¥5 and ¥10 notes. On the machine you've got to pick your origin and destination station.
If you have a single-ride ticket, you need to insert the ticket into the turnstile to enter AND exit the station (so keep your ticket). If you've got Yikatong, you need to tap in AND out of the stations.
The Beijing metro lines generally run between 5am and 11pm. They gets very crowded during rush hours, especially lines 1 and 2. So avoid these times/lines if you can
Yikatong, the reloadable card, can also be used to pay for any Beijing bus, taxi, airport express train, and the metro train to the Great Wall
I recommend getting the Yikatong at the airport express station so you have it before you start exploring the city
The system is very safe. I've never felt sketchy taking the subway and I've taken it late at night
There is cellphone reception system-wide even while underground (not wifi, but actual cellphone reception). This is something I loved about the Beijing metro.
Look for this symbol to find the entrance/exists to the subway stations