My headphones have a 6.35mm jack and my media player has a 3.5mm jack.
If I buy a 6.35mm (female) to 3.5mm (male) headphone jack adapter, will this have any negative effects (sound quality, clarity, etc.)?
Nope, you're fine. Sennheiser, Grado, and the like even supply them with their 6.3mm-terminated headphones.
6.3mm to 3.5mm / 1/4" to 1/8" are super cheap too. Go for it.