CPU: The A8-3850 should approve appropriate for the tasks you plan to use it with. This Processor is the BEST bang for buck, if you're comfortable with its performance. I strongly suggest this for your budget.
GPU: Do not buy a GPU with the APU, it takes away from the value. If you require a GPU, I would suggest something else entirely...
MOBO: Anything that has the features you're looking for. (I prefer ASUS, GIGABYTE or MSI for quality and reliability. They don't have to be expensive either.)
RAM: I know 1866Mhz is tempting, but LOW CAS LATENCY is key. For example, 1866Mhz @ CAS 8 or 9 won't perform as well as 1600Mhz@ CAS 7. That being said, 1866MHz @ CAS7 would 'theoretically' be the best O_O
(Try Corsair or Kingston, maybe G.skill or Patriot)
http://www.tomshardware.com/reviews/amd-a8-3850-llano,2975-6.html <<-- read the bottom section.
HDD: Anything that you like (think about speed / value); can't really recommend anything right now... prices are outrageous. (Western Digital Caviar BLUE)
ODD: Anything that's cheap. (Samsung?)
PSU: Any 80Plus certified PSU will do. I think 380W would be MORE than enough. (Antec, Corsair, Seasonic)
Case: Whatever you like best. There are a lot of decent cases for less than $50. (NZXT Gamma, Rosewill Challenger, Antec 300, Zalman Z9 Plus)
http://www.tomshardware.com/reviews/antec-three-hundred-enermax-staray-cooler-master-elite-430,2707.html
http://www.tomshardware.com/reviews/nzxt-gamma-rosewill-challenger-silverstone-precision,2723.html
You are good to go. I will post another build shortly which costs me less than $400. I think you'll be impressed.