Why do you say 'mon expérience de' and not 'mon expérience à'? When do you use 'à' and when do you use 'de'?
Quite simply because à is primarily (as its root sense, before it began "diluting") a locative preposition that is far more commonly found connecting two verbs than two nouns, and when connecting nouns, it's usually because the expression derives from a verbal one.
You would not write mon expérience à unless you specifically mean experience acquired in a certain place as opposed to your global experience of an object/event/place.
I'm also still a student so this might not be the most reliable advice, nonetheless, you're asking a good a good question, the two prepositions are often confused by most students. They can both vary significantly based on context.
Here are some examples:
*de is normally applied in instances involving the from or those including of,
- Je viens de la boulangerie (I'm coming from the bakery)
- Lucas est un de mes amis (Lucas is a friend of mine)
Whereas à is usually used in instances involving: to, at or in propositions.
i.e. 1. Je suis à la maison (I'm at home) 2. Je vais à Toulouse (I'm Going to Toulouse) 3. Je vis maintenant à Paris (I now live in Paris)
I guess the best way to understand them better is to familiarize yourself with multiple examples comparing both under different contexts, hope this helps, all the best.