Answer:
The answer is: I really need your tips now that I'm starting college next June.
Step-by-step explanation:
As you can see, after each period there's a follow up on the previous sentence, so this is exactly what I decided to include in the blank, a follow up for the sentence "I would love to pick your brain about teaching". In the follow up, Beth gives a reasons as for why she needs "to pick her brain" and includes another word "tips", there's no need to go further as explaining what career, in what college, etc. as the email itself is very brief and clear among them. The final sentence, doesn't include a follow up, perhaps only at the end something like:
Love,
Beth