If you have a way of silencing your graphics card, then a good one will perform as well as an off board DAC. If you go down the head-fi route there are various DAC/amps available, of which the cheaper ones are also portable.
In casual listening you can get £25 (or less) headphones that can sound very good, such as the Sennheiser HD201 and AKG K44. Jump to £100 ( a good but achievable price for the K271 MkIIs) and you will get slightly better sound and more comfort, which in itself is very important. In audiophile listening you get a bigger more detailed sound with the better headphones.
Lossless is lossless and blind testing does not back up claims of being able to tell which codec is supposedly better. So IMO it makes no difference which one you use.