Looking to build a PC for my Mom

  • Budget - I have around a $500 budget. This will be including a Mouse, Monitor, Speakers and Shipping costs.
  • Country/State - USA, Georgia
  • Preferred Retailer - I would primarily like to use Amazon and Newegg
  • Peripherals- I already have a Keyboard and OS, so I will be needing a Monitor, Mouse, Case, Speakers and Wireless Network Adapter. Ethernet is not an option since the router is in a different room over 50 feet away.
  • Things the PC will be used for - She will primarily be using it to surf the web like watching Youtube videos or playing web games on websites like Facebook or Miniclip, but she also will be playing some downloadable games from Steam or Origin like The Sims 4.
  • Overclocking - I don't think it will be needed.
  • Cooling - I think stock cooling will be perfectly fine since overclocking will not be needed and she will not be running any high performance games.
  • OS - I already have a copy of Windows 8.1 for her.
  • Settings/FPS - I asked her, and she doesn't really care about the highest possible graphical settings. She only cares about it not lagging at all. Low/Medium settings would be perfect.
  • Resolution - 1080p, nothing higher.
  • Type of games - Mostly web games as I said, but some downloadable games like The Sims 4. The Sims 4 would actually probably be the most demanding game she would ever play.
  • Type of monitor - IPS would be nice as long as the price isn't too high. 60hz would be fine also, nothing over the top like 120. 
  • Experience with building a computer - I don't have a lot of experience actually picking out a part list, hence why I am asking here. But I do have some experience building the actual PC. I built my first custom PC for myself about 6 months ago and I was very careful, so careful that it actually took me around 6+ hours. But now I know what I'm doing and it wouldn't take me nearly as long for my 2nd build for my Mom.
  • Notes - Even though my budget is $500, it doesn't have to be that much! I would want it to be as cheap as possible as long as it can accomplish what I and my Mom are looking for. 

I put this build together. http://pcpartpicker.com/p/8PpHQ7 The apu should be enough for the light gaming but I might recomend more ram dempending on how much she will use.The ram should be fast enough to get the a good amount of performance with the apu. Also you may find a cheaper hard drive if she needs less space, but this is the most bang for the buck hard drive so... tweak it to her needs.

http://www.amazon.com/Intel-Computing-Gigabit-i3-3217U-DC3217IYE/dp/B0093LINVK

now you have 250 left over for a little SSD or a 2.5 mechanical hard drive and everything else you want.

How good would the integrated graphics in an i3-4130 instead of an APU be? 

I have never heard of that product until just now.

Would it not be cheaper to build a PC with equivalent or better parts?

i guess you are right. it'd be cheaper to buy an old duo-core system from newegg for like 200 bucks.

That can be done well enough with a Kaveri APU. No video card needed in that case. When budgeting always get a decent PSU, but with no video card you dont need anything north of 400 watts. If you can get decent ram, since there is no GPU, good system ram will impact graphics quite a bit. For storage, if she isn't going to store huge games or an entire library of photos, then get an SSD. If more storage is needed then go for an HDD at the expense of a quicker computer.