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.
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 
Thank you so much, guys. 

One thing that is now not that clear for me, is there any difference between a cheap DAC and an expensive one, except for some noise that could be present in a low-priced ones and amount of outputs? Will they produce the same signal for the amplifier?

I'm an amateur musician myself and also wanted to buy the Steinberg UR12 external sound card to make some recordings on it. But will it also work for converting digital to analog for my sound system as an external DAC and will there be a difference in sound between Steinberg and some $100 DAC?

Just like all the other components in an audio system, DACs will have different sonic signatures.  Some are "warm", some are "neutral", some are "bright", some are more detailed, etc. etc. etc.

At the level your system is currently at those differences are probably going to be less obvious.  The more revealing your system is, the more those differences will be noticeable and the more they will matter.
Dragonfly is a good start.  If you want something that is even used at high-end shows, get the Sony HAP-z1es. Maybe get one used on Audiogon or ebay.

Steve N.
Empirical Audio
One thing that is now not that clear for me, is there any difference between a cheap DAC and an expensive one, except for some noise that could be present in a low-priced ones and amount of outputs? Will they produce the same signal for the amplifier?

Big difference in a $10K DAC and a $200 or $1K DAC.  Less of a difference in $300 and $1K DAC.

Better DAC has lower noise floor, better dynamic range, faster transients, better bass, lower jitter (better imaging and focus).

Steve N.
Empirical Audio
@big greg.
Your system will dictate the level of sensitivity you require.
In your case, I would suggest a Schiit Bifrost Multibit.
Given your wishing to use a recording device, buy the Steinberg and see if it satisfies in your system. Seem to me it is very full function at it's price, not designed only as a  DAC. I would think a simple USB DAC would sound better for home audio.

Thanks for the replies, you really helped me a lot. So, an extenral DAC is a way to go, but in some time I will also buy the Steinberg sound card, it would be interesting to compare them. If someone needs an amateur opinion on a difference between $100 DAC and sound card of the same price, write me a message, and I will share my experience :)
You are welcome. Let us know what you purchase. I am following a bid on a HRT Music Streamer II on E-bay. Now it is under $20 (+$7 shipping).  Great way to bring computer audio to a budget system.