Badge are items that modify your skills to make it stronger.
I cant tell you what badge to go for, you have to research them yourself, also also play style.
Mystic badge
Mystic badge can only be found in Celestial Basin(no longer in dragon express).
Celestial Basin can be unlock by doing a letter quest. Check out After Act 8 guide on Celestial Basin
You either need MSP flower or Raven feather along with peaches to exchange for a badge.
Soul Badge
Soul Badge can be found in two places, Mushin Tower and Dragon express with Solar Energy(current long term event)
Mushin Tower with Token:
You need 70 Token to exchange for a badge.
Tokens can be acquired from doing Tower of Infinity, where the higher you go, the more tokens will be given.
Previous season Token can only be bought while current season Token can only be farm.
Mushin Tower with Prayer Beads:
Eternity and Transcendence Badge can be found here with Yunsang Prayer beads, obtain from F16 to F20 Mushin Tower(NOT Tower of Infinity)
Dragon Express with Solar Orb:
Solar Energy can be obtain doing daily.
Only previous season badge can be exchange with 125 Solar Orb, which will take you around 1 month to get 125.
It is recommended to use this method to get your Soul Badges as certain Tokens can cost a bomb.