How do I get my mom to respect that I'm old enough to have a serious relationship?

Depends. My assumption is that you have some mental, emotional or physical handicap that makes your mom worried that your involvement in a relationship would affect your wellbeing. If this is true , you need to make her feel that you can be safe in that relationship. You have to prove your maturity to earn that respect.
Clear communication. The trick is to demonstrate your ability to be responsible, calm, non impulsive, and to share your thinking and intentions. Show that you are patient and that you are willing to keep her "in the loop, " so that she can see and feel that you are not being silly, irresponsible, over-eager, and that you want to transform to a new, adult-to-adult relationship with her - one of respect and appreciation.