Newly licensed HAM here. Are QSL cards only exchanged during DX, or could they be exchanged with local contacts as well? How does one request a card? I’ve heard there are bureaus that route cards; is this the most appropriate way to send a card?
They can be sent to anyone you've had a QSO with, whether it's that DX contact or the person next door. You can send them directly through the mail, or through the outgoing QSL bureau. Generally, you'd want to use the QSL bureau if you are sending out a large batch of cards, to save on postage. You can request a card just by asking when you make the conact, or send out a card and wait for one to come back (usually you have something like "QSL PSE" (QSL please) on your card that you mark or circle indicating that you'd like a QSL card in response)
You may also wish to consider electronic QSL sites like eQSL and Logbook of the World. These types of sites allow QSL information to be automatically exchanged between stations. In many cases, they are directly integrated into the ham's logging program.
While not as visually appealing as a printed QSL card, these nearly instantaneous electronic QSLs are virtually free and they can be used for many of the popular awards if you get into "paper (award) chasing". You can still send paper QSL cards through the mail but due to ever increasing postal costs, many hams have gone fully electronic.
Another service you may wish to consider is Global QSL. They print on demand a double sided, color card filled out with the QSO details and then mail it for you. You can manage 10 different profiles through their site and change card graphics at will.