I never understand why so many people don't specify a budget for these "Which ____ should I buy?" questions.
You have tons of options. There are $300 powered speaker solutions like the very popular Audioengine models.
Do you want wired or wireless? Do the controls matter (like a conveniently placed volume knob)? Do you want to plug anything else into the speakers? What size speakers would be the maximum you would like?
There are junky "PC speakers" for $30. Pretty much everything will sound better than the Mac mini's internal speaker so your budget is an important factor here.
For what it's worth, I am using a discontinued Teac AI-101DA USB DAC amplifier, a pair of discontinued Polk Audio TL1 satellite speakers, and a long-discontinued geriatric Cambridge Soundworks BassCube 85 subwoofer. The latter two are probably fifteen years old.
The Teac amp is maybe five years old, has inputs for USB, Bluetooth, two optical digital inputs, and analog 3.5mm mini-jack line-in (and subwoofer out). The Teac amp's DAC can convert 192kHz/24-bit digital audio to analog. Pretty decent output from a tiny amp.
If I listen to lossless music (mostly classical music, opera, some jazz), the speakers are the limiting factor not the amp. Videogame audio is also pretty awesome especially with the powered subwoofer; over-the-ear headphones hurt my head after 30 minutes anyhow.
If you're mostly watching YouTube, the audio quality is likely terrible and whatever speaker you use probably won't make much of a difference unless you are sticking with content with extremely high, professional grade production values (which is probably <0.1% of all YouTube content).
I did accomplish what you are setting out to do: a clean looking, compact desktop speaker system for a Mac mini. The amp's circular volume knob is within arm's reach; I prefer simple physical controls for audio volume rather than buttons or a graphically-based slider. My desktop is maybe 3.5 ft. by 2.5 ft. so I didn't want large speakers (the Polk TL1 is 6.5" tall). That cripples bass performance so the old subwoofer in the footwell makes up for much of the low frequency shortcomings of the TL1s.
Could you accomplish this with a Harmon Kardon SoundSticks III or SoundSticks 4 system? Absolutely, although the volume controls are on the subwoofer, an inconvenience for some.
