Exploring an existing network

I am very new to networking.  Walking into an existing network; the administrators who set up the network have vanished, leaving not a trace of documentation.  I am a programmer by trade, need to learn networking in a hurry.  Happily, the network is functioning OK, at this point just need to connect a new printer, stuff like that.  But eventually I will need to actually do something hard, and before the crisis hits I'd like to get a good understanding of the network.

I figured I should start by knowing what the pieces are.  I made an elementary map of the connections that I can see (e.g., the ones not in the wall). Basically there is the cable gateway, and there are two routers, they are what I would call daisy chained, not sure if that's correct terminology.  There are a few peripherals, one server, four hard-wired workstations, one wireless laptop.  Is it useful to know what's connected to what, or am I just being OCD about my pretty drawing?   Is there a tool that I can use on the server to definitively find out what is connected to what?  

And how can I tell what security is in place?  Is that administered through an OS level admin tool, or is that done at the routers?  The routers are Linksys.

I'd like to be able to do administration remotely; is Remote Desktop to the server the best way to do that?

What sort of basic maintenance or watchdogging should I be doing in general to make sure things continue to go smoothly?

Is there a good webpage where I can go to get a good understanding of all of this, or a book that someone can recommend?  Obviously I have a lot to learn and need to get up to speed quickly.  So I'd prefer to learn what applies to my current situation, without too much interesting but not applicable information . . . :-)

(I know this is a lot of questions in one, but it seemed silly to re-explain the situation and post each question separately.  If one in particular is a big question and should be separated out, let me know.)