When you say Coaxial, do you mean optical in/out, RCA cables, or digital coax? I have never seen a motherboard that comes standard with a coaxial out (or digital coax mostly for home theater applications). Most come with digital in/out (red laser), however.
If you will be using your computer, I only see two practical ways to go: with USB -> DAC -> AMP ->Headphone OR dedicated internal(or external, actually) sound card with DAC -> AMP (or straight to headphone using the sound card's amp) -> Headphone.
I never had a chance to compare the sound quality between a nice sound card and USB/DAC but I would think they wouldn't make much difference if at all, based on science.
Digital in/out, digital coax, and USB are all a digital source (1's and 0's) just like CDs so they all should output exactly the same info. It's the receiving end (DAC) that matters which processes the digital signal into analog so you can hear the music. RCA cables (red, white) are analog.
In any case, on-board sound (embedded on the motherboard) sucks just like all on-board Intel video sucks big time. This is why people pay upwards of $500 for premium video cards so they can play the latest games with all the bells and whistles.
If you play games and want to take advantage of all the premium sound card has to offer (surround sound) and also want to listen to music at the same time, I would go with a nice add-on sound card.
If music is the only thing you care about, I would go with a separate DAC/AMP since you can get a decent one for the amount of a nice sound card but also get to use it for other setups if needed (like E7 being portable).