Pass Without Trace gives the player a +10 bonus to stealth checks. Would I add 10 to my initial stealth modifier or swap it for +10?
If I had a +2 stealth mod and use the spell would I have +12 or +10?
It's a bonus, and generally, bonuses stack, so you add it to your current Dexterity (Stealth) modifier. Much like how you add your Proficiency bonus to your Dexterity (stealth) modifier if you are proficient in stealth.