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Yes I am currently using the on-board audio and I don't mind getting a new sound card or DAC but is it better to get a sound card or DAC then a 1/4 to 1/8 adapter.
If all your really into is music, then going external might be a better choice.
With external audio hardware, any electrical noise inside the computer case will not effect the audio quality.
(if there is electrical noise inside the computer).
With external you can spend hundreds of dollars ($300, $500, $1000, etc.), for a DAC and amp, for some great audio quality.
If you need Headphone Surround Sound for games or movies, it's better to get an internal sound card.
The most you can really spend for internal sound card is $250.
If you get an external DAC with S/PDIF (optical or coaxial) input and your motherboard has a matching S/PDIF connector,
then you can go all external and still be able to use the motherboard's built in audio features.
You can also go with a cheaper sound card $50-$100 (with built in DAC chip) and get an external headphone amplifier to connect to it.
Between 1/4" or 1/8", it makes no difference for audio quality, use an adapter or not.