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)?

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.