Live-video beginner, where to start?
There are two main concepts that I think are key here.  The first is learning about signal paths and what the functions of each piece of equipment does along the way.  The second is understanding the signals themselves and how they work.  In video, this can be particularly important since many signals are actually the same on different cables.  In both cases, I think the best way to get started is to research each individual device and signal.

For signal flow, simply looking at how things are wired up should help you start.  You've normally got something similar to a camera that runs in to either a video switcher or a time base corrector followed by a switcher.  From the switcher, there are feeds to the preview and production monitors as well as an output that most likely goes to a Distribution Amplifier and from there it goes to the outputs which are probably projectors.

If you understand the purpose of each device, that is the first step towards understanding where problems may be coming from.  For example if one camera's signal is bad but the others are fine, check the camera and cable.  If all the output has the same problem, it may be the switcher or something between the switcher and DA (or the DA itself).  If only one projector has a problem, then either the DA, the cable from the DA to the projector or the projector itself has a problem.

The second part is understanding signals.  This is more in-depth and many video guys don't get this deep in to things.  Again, Wikipedia searches on either the cable or signal type are a great way to get detailed, accessible information.  To talk about the ones you specifically mentioned.  NTSC and PAL are actual video standards.  NTSC and the Japanese version NTSC-J are 29.97 frame per second interlaced (every other scan line is updated) formats.  PAL is the equivalent of NTSC that the rest of the world uses and is 25 frames per second, interlaced.  They are both analog, standard definition formats.  

BNC is a type of coax cable connection.  It is a twist lock connector that can attach the cable securely and is generally used for professional video.  It also used to be used for old school Token Ring networking (on computers).  BNC cables can carry a very wide variety of signal types including Composite, Component and SDI.  BNC refers to the physical medium and not the format of signal passing through it.

DMX is a lighting control protocol that works over either an XLR or a 5 conductor wire similar in connection to XLR cable (but with 5 pins rather than 3) or a standard RJ45 cable (network cable).  There are also various wireless implementations of DMX.  It supports 512 addressable channels with 256 possible values per channel.  It can be used for controlling either robotic or fixed lighting.  With fixed lights, one or two channels is used per light. (Dimmer and possibly color selection)  For robotic lights, any number of channels may be used and each channel controls some different value such as gobo selection, x axis movement, y axis movement, strobe settings, dimmer, color, prisms, etc.  Additionally, some devices may use 2 channels to give finer control over a single axis.

There is also far more information available on Wikipedia in a fairly friendly and accessible form assuming you have a fairly basic understanding of electronic concepts.  That is how I have picked up the majority of my knowledge.  You can also learn fun things like the fact that VGA is identical to the analog feed in DVI which is also a component RGBVU feed.  Also, the digital signal on a DVI-D cable is the same as an unencrypted digital signal on HDMI.

Identifying the keywords to search for is probably the biggest trick, but if you know the cable name for example, you can likely find what the signal type is.  Also, if you ever end up stuck on a particular signal or cable, feel free to come back here and post a question and someone would be happy to point you in the right direction and provide you with details about the particular signal or device.

Oh, one other term that might be worth searching is Gen-lock.  It's a clock signal that is how the frames of video signals from higher end cameras are synchronized. Most modern video switchers can correct for time base shifting automatically now, but for minimum lag in display, if the video cameras support gen-lock, then using it to have them sample together can be helpful.