How can I repair scratches on a painted French door?

A dog chewed up our solid wood, painted french doors. The tricky part is he got the trim between the glass panes and along the edge where the doors meet. If we want to repair ourselves can someone tell me how much time this could take? What materials I need and how much it will cost? If in your opinion this is a repair for a professional, what type of craftsperson should I look for (carpentry?), and how much would you estimate the cost to be? Thank you for your time.

update: 6/21/19 1:42 PM CT

Thank you so much for the quick replies! In addition to the methods and tools to use, does anyone have an estimate on the time it would take to fix this door and how much it would cost? The dog's owner gave us a $100 gift card to Lowes, without asking how much the damage cost. It's not really the money I'm concerned about. I just don't know if I have the talent to fix this, and I certainly don't have the time. I'm not sure what to do.