How do I know I'm not drawing too much current for my 3.3V supply?
The largest value I can find in the datasheet for the ARM is 126ma when running flat out and peripherals on. This is unlikely to cause problems. If you're very worried you can buy desktop power supplies with current limiters. At the very least, when powering it up for the first time use a phone charger not a computer and have a suitably-ranged current meter in the positive line.

Are you intending to connect the ARM to the USB? There will be a reference circuit for that - use it. That bit also requires careful PCB design.

I would start with a reference design for the Stellaris - there's a cheap dev board available for it - and work through understanding why everything is there. If you decide you don't need it, remove it.

(These parts aren't as forgiving and easy to use as PICs / AVRs, and therefore less likely to produce a working circuit)