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)?

The amp does not appear to have a digital input.

What I would suggest is you start with the PC's speaker/headphone outs to your amp first. At least play some music!

After this .... I prefer USB output, but it can be complicated. If you have a rear S/PDIF connector that will expand your possibilities to getting an older DAC without a USB input.

Please correct me if I'm wrong, but I though that a DAC/sound card will transorm digital signal from PC to analog and then I could connect DAC/sound card to an amp through a stereo cable. Is it wrong?
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.