Microchip was kind enough to send me a few sample Atmega328 chips
Snort! Not very likely! 
(You probably mean "Atmel"? Microchip is sorta their arch-rival!)
My first question: What is the best board to get started with? There are a ton of arduino clones that are a bit cheaper. ... I am a very poor college student, so things like the Solarbotics Ardweeney are quite tempting due to their 9.99 price.
Well, given what you've said, I would skip the cheaper clones that don't allow you to use a socketed ATmega chip. No Seeeduino or Arduino Pro because they use SMT chips. No Ardweeney because it solders directly to the ATmega. The kits from MDC are nice. I'm "cheap" but not poor, and my initial Arduino-related order was for an MDC BBB kit, 5 BBB Bare boards, 10 pre-programmed chips, and an FTDI cable. Then again, I later came to realize that I was a bit limited by having a "non-standard" board, and ended up acquiring an official duemilanove as well. (I mean, the very first BLINK sketch "doesn't work" on a BBB, because there is no pin-13 LED...)
Can I use the cable I already have (Amazon.com...)
Or will I have to get one of these breakout boards?
You should be able to use that cable. See DIY USB to Serial Cable For $3! – uCHobby and/or search the forums for "CA-42"
What supporting hardware do I need to get an arduino up and running on its own separate from the Duemilanove board?
You need a crystal and caps (or a ceramic resonator), some bypass caps, and a pullup resistor for the RESET signal. I highly recommend some sort of PCB that implements the basic wiring and a socket; at about $5 it's worth the certainty of a working hardware design.