Is elemental damage listed somewhere?
There is no separate line for elemental damage in your character sheet.
You are correct that the skill shown in your character sheet deals no fire damage. Note that your character sheet has all your active skills listed. Double check that you are looking at stats for your primary damage skill, not the default attack.
Your new weapon likely deals fire damage. Fire damage numbers will be displayed in red (colour code for fire), just as cold damage will be blueish and lightning will be yellow. It does not mean your fire damage will be lower.