A quick guide
If you're going to use the metro (or busses) a lot, buying a metro card is a good investment. As of August 2018, you can only buy and refill your card with cash. The cards are sold in separate vending machines, but they're usually right next to the refill machines.
To see your card's balance, you can at any time place your metro card in the vending machine. After a second or two, the machine will take you to the refill menu and in the top right corner is your current balance.
5,000 won will buy you at most 4 trips (prices as of August 2018). The minimum fare is 1,250 won.
In the green (line 2) city circle line, inner city circle runs clockwise, and outer city circle runs counter clockwise.
For most lines you just need to know the terminus, to choose which side of the tracks aim for. But the signage for the green line (line 2, city circle) is based on pointing you in direction, based on major "nearby" stations. So, you basically have to know at least one major station between your current station and your destination. For instance, if you're going from City Hall to Hanyang University, the signage for the right side of the tracks, will show stations such as Wangsimni and Seongsu.
Another thing to know about line 2, is that Google Maps always says Seongsu as your terminus/direction. This is not necessarily the best way for you to go. Revert to using the aforementioned methodology of knowing major stations between your current stop and destination.
Almost every train has a map of the line and a map of the entire network. The one with just the line shows for each station if you exit on your left or right (according to driving direction). If the station's marker dot is yellow you exit on your left-hand side, white is your right-hand side. In rush hour, it might be an advantage to position yourself on the correct side as early as possible.
On Google Maps, if you zoom far enough in, you can see the metro station exists, as numbered yellow circles. Be mindful of the fact that you may need to zoom further in or out to see certain exists. It will rarely display all at once, for some reason. You can usually follow half-way transparent lines that are the same color as the metro line to find the exits. Find and remember the best exit to get to your location. When you exit the train, follow the yellow signs pointing to that number.
Exits are also convient places to agree to meet up with friends.
Google Maps is also great for finding your route. Even though general navigation is not available, it can still search in public transport. Simply input your departure and destination, and choose the icon for public transport.
Downloading an offline map is a good idea. It's not always easy or convenient to use the one on the train.
The few seats that have a different color are reserved for the elder, pregnant and handicapped. It's considered disrespectful to use them, even if almost nobody is on the train.