Im a (very) casual KiCad user and currently its been a few months since I have really used it, so I might not remember everything exactly or have too good advice to give. But yes, I like it very much and its my software of choice. In fact I started using it simply because I needed some PCBs that were 10 cm squared a couple of years ago.
Fortunately there are some good tutorials around, like the one you found. I also like this one from curiousinventor:http://store.curiousinventor.com/guides/kicad/
That got me started on some things that were baffling at first. Because KiCad can be somewhat quirky to work with, I guess mostly because it is "mode based" as you say (that you are in "move objects" mode, or "connect wire" mode, etc), but also other stuff like making custom library components/symbols and footprints ("modules").
Btw one thing I like about it (KiCad), is that you can mix and match components with custom footprints on a per-project base.
There is even a page with libraries converted from Eagle here: http://library.oshec.org/
I have to say for my modest needs I havent used most of those (I think I used 7400 library, at least for a template for modifying a few of my own).
Probably lots of pages with libraries here and there, here is another one with misc components and their 3D shapes: http://www.kicadlib.org/Fichiers/Ludek_libraries.html
I have also made a couple here: http://raronoff.wordpress.com/2010/08/08/a-few-random-kicad-components-rgb-led-stuff-mostly/
Personally I use more than one computer, and more than one OS (but I favor Ubuntu for working on my stuff), and one thing that annoyed me a bit was the need to manually re-add the libraries used in any given project, if you suddenly find yourself at another computer for a while. Yes, there is a "user defined search path" list or setting, but I havent had much luck with that (newer versions of KiCad might have fixed this?). Anyway I found a solution that worked for me using symbolic links, but it would have restrictions on (probably) one user pr. KiCad innstallation (not disregarding the possibility of installing KiCad more than once, say once pr. user), and possibly not across network drives (not tested).
Anyway, I wrote more about that here (feel free to just disregard it, its not essential for using KiCad): http://raronoff.wordpress.com/2011/03/19/kicad-cross-platform-hassle-free-project-sharing/
If you keep both EEschema and PCBnew open at the same time, and click on a pin, the corresponding pin and track will be highlighted in the other program (well might be a bit hard to see the pointer in EEschema, as its not really highlighted there). Sometimes handy, especially track highlighting.
KiCad can also export the net to an online router http://www.freerouting.net/
As for autorouting I dont know how good that is, but it does have a nifty track pushing that repells other tracks, among other things (based on me trying it once, a while ago). I usually route things myself, but I dont have any complicated projects yet (and still lots to learn).
Heh, while writing this, I have learned that KiCad (build 2011-03-18 BZR 2903), finally have rotate block in both EEschema and PCBnew. And also an export VRML (.wrl) option for 3D shapes for the design. Nice! For windows at least, the Ubuntu repositories is a bit lagging unfortunately.