Advice needed: am I going in the right direction?

I'm new to hi-fi sound world and spontaneously bought myself Yamaha NS BP200 passive speakers and Onkyo A-9030 amplifier, because I thought that connecting it to a PC will be a piece of cake - just buy some external sound card and you are good to go!

However after doing some research I found out that it's not that simple and now I'm kinda lost: does it (PC -> DAC or sound card -> amp -> speakers) even work like this. So, my question is:

Am I going in the right direction? And if yes, maybe you can suggest some DAC/external USB sound card that could suit my case (budget is about $150)?

That amp has multiple RCA inputs. 

You run a USB cable from your computer to a DAC.  The DAC will have RCA outs that you can connect to one of the RCA inputs. 

You don't need digital inputs on the amp.  Not complicated at all. 

I don't have any recommendations for a DAC in that price range, but I know there are lots of choices available. 
Your PC may even have an optical out, which I think is better than USB- though some may disagree.
But, big greg is correct, use a USB or Optical cable to the DAC of your choice. The rest is like the usual stereo connections.
As for a sound card, I wouldn't bother. Just use the PC to DAC.
There is a program that doesn't cost much called Fidelizer. It really made a difference for the better.
For a good inexpensive DAC, I would get a Schiit Multibit or Gungnir.
They are available used at a great price, too. Just make sure it is using the current most current updates. But, even if it isn't you can send it in to be upgraded. Great company.
I would get an Audioquest Dragonfly.  It is a DAC that looks like a usb stick and fits your budget.  You will need a cable that has a mini jack to fit into the DAC and terminates in RCA inputs for your amp
You are correct. Connect computer to DAC, DAC to amp, amp to speakers. Computer needs to recognize DAC, may need a driver depending on DAC and type computer. 
If using USB out from computer and you want to keep it simple and inexpensive there are HRT Music Streamer DACs for sale on Ebay, actually one listed for $57($5 shipping) that would get you started. I have used these in several budget systems and they worked well. Another option is the Schiit Audio Modi 3 which would provide more input options for around $100 shipped.

Good luck