I recently purchased a new computer (parts) from Newegg. My motherboard is an ASRock P45XE,
http://www.tomshardware.com/reviews/intel-p45-core,2110-2.html
http://www.newegg.com/Product/Product.aspx?Item=N82E16813157143
My processor is an E8400, and I have 4GB of G.Skill DDR2 1066.

In the BIOS settings, I change the FSB of my processor from 333 to 356. Which is 3.0Ghz to 3.21Ghz. Increasing this, changes my DRAM frequency from 1066 to 1138. For some reason, my computer won't function after the BIOS screen with 3.21Ghz AND DDR2 1138.

Although, I am able to run my computer safely at 3.21Ghz and DDR2 854.

So my question is... Should I run my computer at 3.0Ghz and DDR2 1066? Or should I run it at 3.21Ghz and DDR2 854? For the best performance

You can OC and manually set a divider in the BIOS to allow you to run your RAM and FSB at different speeds. For your given numbers, you can set a RAM:FSB divider of 8:5.

Your OC may also be limited by your motherboard. But generally speaking, a higher CPU frequency nets a greater boost in performance compared to a higher RAM frequency.