How to get to the city from Beijing Capital International Airport (PEK) cheaply by train

Taxi from the PEK airport cost ¥100-¥130 ($15-$19) from my experience. While it's faster than the metro, during morning and evening rush hours it takes about the same time as the metro
If you don't have a lot of baggage with you, I recommend taking the metro train called Airport Express to get into town. It's much cheaper at just ¥25 ($3.6) each way, and it's connected directly with the rest of the metro system that can get you to anywhere in Beijing
It runs very frequently at once every 8-10 minutes. First train leaves terminal 2 (T2) at 6:36am and last train leaves T2 at 11:10pm everyday. See the current schedule on the official site
It departs from T2, then goes to T3, then Sanyuanqiao Station, then finally Dongzhimen Station, from where you can hop onto line 2 of the main Beijing subway system. The trip takes 20 minutes from T2 to Dongzhimen, and 30 minutes from T3 to Dongzhimen
If your flight lands in T1 (which is only domestic flights), you'll need to take a free shuttle bus to T2 or T3 to catch the Airport Express
How to take the Airport Express train:

Follow signs inside T2 or T3 to get to the Airport Express station. The sign should look something like this:
Buy a single ticket, or I recommend getting the reloadable metro card called Yikatong (一卡通) that you can use anywhere in Beijing for subway, bus, taxis
Take the train to Dongzhimen subway station. From Dongzhimen station, you can transfer to anywhere in Beijing on the subway. You don't need to buy another ticket or do anything special at Dongzhimen station. You just need to insert your single ticket or tap your Yikatong card at your final destination station when you exit

Other tips:
I used this app/website to help figure out how to navigate the Beijing metro system to get to the final station. Google Maps is also good to see what is the final station I should get to that's close to my hotel
If you have a lot of luggage, you can also take a taxi at Dongzhimen Station or Sanyuanqiao Station instead of transferring to the main subway system. Because you're already close to the city center, the taxi will be much cheaper from here than from the airport. For example, the taxi from Dongzhimen to Tiananmen Square is only around ¥30 ($4.5).
If you want to catch a taxi, I actually recommend catching it at Sanyuanqiao Station instead of Dongzhimen Statino. It's less crowded and you'll spend less time waiting for taxi
There will be escalators at these stations so you don't have to worry about lugging your bags upstairs

Let me add a related brief