Also remember if you build it yourself you lose the warranty that someone like HP, Gateway, Dell, iBuyPower, etc. will provide.
You'll have to tech the pc yourself or find someone who can fix the problem.
The integrated sound chip usually works fine. It's what I use for gaming.
For integrated graphics, IIRC you'll need to get a sandy bridge cpu that supports it. For example, the Sandy Bridge i5/7 2500k/2600k doesn't support integrated graphics.
But I believe the Sandy Bridge i5/7 2500/2600 does support integrated graphics.
I think buying a separate video card or using integrated graphics depends on what games you'll be playing.
Hardware performance - Minecraft Wiki
Most of those video cards are rather cheap compared to the newer $300 cards.
Main components
Tower (if needed [make sure your new motherboard will fit your old tower if you're using an old tower])
Fans (if not included with tower)
Motherboard
CPU
CPU Fan (if needed, CPU should come with a standard fan though)
PSU
Video Card(if needed)
Sound Card(if wanted/needed) I personally use the integrated sound
DVD(bluray) writer/reader
Hard drive
Solid State Drive (optional but I'd highly recommend)
Thermal Compound
Keyboard/Mouse (if needed)
Operating System (Windows/Linux [ Windows IMO ])
Screwdriver with phillips/flat head
Patience
If I were you, and you haven't already, I'd sign up for the newegg newsletter to get their daily/weekend/special deals.
Then when a good deal hits you can get buy the component you need and save a few $.
Sometimes you can grab a good combo also.
Example(From daily deal):
Newegg.com - Shell Shocker Deal. Exclusive Jaw Dropping Savings on PC Components and Electronics.
Newegg.com - Microsoft Windows 7 Home Premium SP1 32-bit - Operating Systems $99 minus $10 savings with Promo: EMCKAGK33
Newegg.com - Official Newegg Promo Codes, Coupon Code, Coupons, Discounts, Promotions, Free Shipping Codes