last name question

so my baby will be given both mine and his dad's last name (we aren't together). but because I'll be the one that deals with school and all the paper work when he grows up I want him to primarily go by my last name, just to avoid confusion. my question is should I put my last name first or after the dad's? I always figured it should go first but I had a conversation yesterday with someone who said mine should go last because people will use the very last name as his 'last' name. but idk! plus it doesn't sound good with my last name last.. and his name will be very long already 🙄 any advice? thanks in advance