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Logs
rpierce99 actually made a great app called GetLogs. Check it out, makes taking logs much, much easier! (Does things like auto-masking, direct posting to pastebin, etc.)
If you want to pull logs manually, the most popular kind are dmesg, logcat and logcat -b radio.
adb shell dmesg > dmesg.txt will dump a dmesg log (kernel traces) to a file — dmesg.txt — on your computer. Note this command will drop dmesg.txt where ever you are currently running the command from — watch the path you are at!
adb logcat > logcat.txt will dump a logcat file (userland traces).
adb logcat -b radio > logcat_radio.txt will dump a logcat -b radio file (logcat specific to the radio/RIL).
You can also run these on the phone directly. Note: dmesg runs and ends while the two logcat commands run continuously. You will have to break them (CTRL-C) to get the log to quit and give you back a prompt.