Making your own processor board can be a lot of fun, recently I have made a few different ones, but I ended up doing this by having the boards to be fabricated and then I did all of the assembly work. This was a lot of work as I used mostly surface mount components and I added a lot of things, like switches, speaker, power buss, ... If you are going to build your own boards you should invest in a nice soldering iron/station, plus magnifying light. If your board uses all through hole components, then while I would still recommend getting a good quality soldering iron/station, it does not need to be as good.
Something I would also recommend getting is some form of multimeter, that you can measure volts and ohms. If something does not work, this helps out to figure out where you may have a short or where you may have an open circuit.
However, I am not sure if I would personally recommend this as what I would do for my first project. It really depends on your interests. Personally I think Purchasing a kit can also be a good way to start off as there are often lots of other people that have also built the system, and they often can help out when there are issues. Also they can usually get you up and running quickly enough for you to experiment and find out where your interests lie. Are you more interested in the electronics? Or the hardware? Or the programming? Also depending you where you live, there may be clubs around that have workshops that you can attend and learn. Example: http://www.seattlerobotics.org/WorkshopRobot/index.php.
As for Arduinos versus other boards? Again hard to answer. The plus to them are that there is a lot of support for them and lots of different boards you can purchase for a cheap price. Downside is that the boards don't typically have any standard servo type connectors and the associated power buss... The boards that are sold here are nice for that. There are also some other boards out there that added this plus still maintain compatibility with Arduinos, like(http://www.lynxmotion.com/c-153-botboarduino.aspx
). The Arduino environment is free, but others prefer to use standard C or C++ without the Arduino stuff. I agree with them, but currently do most of my stuff within the Arduino environment as it is fun to share...
Most of my robots come from Lynxmotion, but I do have one from Trossen and have had others, like Robonova. So forgive me that I mention mainly Lynxmotion products.
One thing you will find when you start playing with Robots, is there is lots of different types of robots. Sometimes you have to experiment to find out where your interests lie. Examples:
a) Wheeled robots. Like the $50 here, or the SRS one I mentioned above, or one from Parallax (http://www.parallax.com/Store/Education/KitsandBoards/tabid/182/ProductID/820/List/0/Default.aspx?SortField=ProductName,ProductName
). These are nice table top learning robots. If you find you want want to get off the table and more speed or load and expandability (but at a higher price), I have had a lot of fun with the Rover from Lynxmotion: http://www.lynxmotion.com/p-860-a4wd1-combo-kit-for-autonomous-botboarduino.aspx
b) Or you might like track drive robots...
c) Or you might like Biped walking robots. There are several of these out there, like the Robonova, or Robotis. One of the simplest ones is the Lynxmotion Brat: http://www.lynxmotion.com/c-97-brat.aspx
d) Or you might like multi-leg robots like hexapods. This is where I find I have the most fun and have several, including: My first one from Lynxmotion: http://www.lynxmotion.com/c-101-ch3-r.aspx
and my most recent one the PhantomX from Trossen: http://www.trossenrobotics.com/c/phantomx-hexapod-kits.aspx.
But as these all have at least 18 servos (some have 24), the cost goes up...
Again I am only touching the tip of the iceberg here... Sorry for my long winded answer