I've been asked to put some of the resources I use for my projects all in one place. Here are a few of my favorite things. I regularly try to do video demos of my projects, and so for that take a look at my youtube channel. Also, you can look at sites like hackaday for ideas, or even your local Buffalo Lab makerspace.
The main Arduino site itself is obvious, but I also like Sparkfun and Adafruit as both sources of ideas and for things like shields. If you are going to use the USB Shield or the Mega ADK board, the Circuits@Home site is very useful as well. Also, don't forget about the "examples" that are available within the Arduino IDE itself.
If you are planning to skip the Arduino IDE and "sketch" system completely and build code like your forefathers, you need to understand the underlying chipset and have a working toolchain. It is very instructive to write a simple C program, have it be compiled to assembly, and to try to discern what that code does. You get extra points if you grok relocatable code and understand what a linker does. When you finally have object code to upload to the device, you need to choose how to do that last step. Fortunately, the Arduino hardware has a built in method of flashing the firmware via the USB port. You can get an AVR programmer if you want to be hard core, or you have a board that doesn't have the secondary bootloader.
When I'm doing electronic design work, I usually pull parts from places like Jameco and Digikey. there are others, but between these two you can find almost anything.
Random tip: Always be aware of what voltage logic you're using. For example, most Arduino boards are 5v, which means they output 5v as a "1" and they have a fairly high voltage threshold for inputs to be considered a "1" (I think something like 2.7v). If you're trying to interface with a device that is based on 3.3v logic or even lower, you can get odd results or even damage your devices. There are ways to do level shifting using options as simple as a voltage divider.
There are some standard ways to do things like light an LED from an output pin or "debounce" a switch that is attached to an input. Understand how these simple circuits work.
The developer site is a great resource, as is all of the free code on Github (including my stuff). If you need help, a google search usually turns something up on stack overflow.
Hope that helps some. Let me know if I missed anything useful.