こんにちは (Konnichiwa) EarleyGrave,
Thanks for your question.
You're absolutely right! You can translate え (e) as "painting," "drawing" or "picture" and りょうり (ryōri) as "cooking" or "cuisine."
え (E) can mean "painting" or "drawing" as in a completed piece of art, but it can also mean "drawing" or "painting," as in the art of making a drawing as a skill. Here it talks about this activity as a NOUN. If we want to talk about "drawing" or "painting" as a verb, we can use え を かく (e o kaku).
りょうり (Ryōri) means "cooking" as in the art of cooking as a skill, not the verb "to cook." This means it is also a NOUN. The verb for "to cook" is りょうり する (ryōri suru).
Since we can only use が じょうず (ga jōzu) with nouns, we need to use the noun forms here.
So what this sentence is really saying is:
"I am good at (both the art of) drawing and (the art of) cooking."
If you want to talk about the action (verb) of cooking and drawing specifically, you can use の が じょうず (no ga jōzu) instead. You can learn all about how to use this grammar in this lesson:
I hope this helped! Please let me know if you have any other questions.
べんきょう を がんばって ください！ (Benkyō o ganbatte kudasai!)