For recording, mixing and producing, we want a PC that is powerful enough to run any DAWs/Plugins, quiet and doesn’t require too much money (because money is needed for the audio equipment too).

Mid-range is ideal.  Don’t go overkill. 

You want something more powerful than your average laptop for sure, but you definitely don’t need the best of the best.  High-end may have once been required, but technology has been catching up and DAW’s/Plugins haven’t become that much more demanding.  Nowadays, a mid-high end laptop and any PC with an Intel Core i desktop processor from the past few years is plenty.

Is a video card needed?
No; not at all.  CPU with integrated graphics can handle plugins and DAW’s just fine.  However, if you also plan on doing video editing (or some gaming for fun) on the machine, a video card is good.

Recommended CPU’s (as of Feb 2018):
i3-8100, i3-8350k, i5-7400 and i5-7600k are solid choices.  The Pentium G4 series is not necessarily ideal, but is a great value for those on a very tight budget. 

Ryzen 3 and Ryzen 5 are recommendable to those already planning on using video cards, but they lack integrated graphics.

There is generally no reason to spend more than ~$200 on a CPU.  An i7 or Ryzen 7 is honestly total overkill.

How much RAM?
8 GB minimum.  16 GB ideal.  8 should be fine for many.  Try out 8 and see how you like it before upgrading.  Above 16 is overkill.  Brand doesn’t matter at all as long as it’s a reliable company.

What storage should I use?
A solid state drive is most recommended due to overall speed of loading instrumentals/sound files, opening programs and booting up in general.  A 120 GB SSD is better than none at all, but if you use a lot of plugins and instrumentals, you’ll want a 500 GB SSD.  Using a plain old hard drive isn’t the end of the world, but SSD’s are recommended.  Brand doesn’t matter at all as long as it’s a reliable company.

What monitor should I use? 
You’ll want enough screen size and resolution to allow for ample workspace.  Some may prefer to have 2 22-24″ 1080p monitors for multitasking.  Others may want one larger monitor with higher resolution.  4K is an option for some, but it’s expensive, some programs scale poorly on it.  For audio work, my overall recommendation (best bang for buck) is 27″ 1440p.  IPS panels are recommended.

How do I make my build quiet?
Read this article. 

Other recommended parts?
CPU Cooler: Be Quiet Pure Rock Slim
Case Fans: Be Quiet Pure Wings 2 
Case: Fractal Design Define (Mini C for MATX ; Nano S for ITX)