Taking courses at the LSA Institute offers an unparalleled opportunity for exposure to a vast array of courses, professors, and students. If you are able to swing the cost, I highly recommend it (there are also very competitive scholarships — always worth a try!).
To be sure, it’s one of the best ‘networking’ opportunities a hopeful student has available to them. Most Summer schools only last for a week or two; the LSA Institute is a whole month. In my case (a few years ago), I made tons of friends (students and professors) which have lasted, and new friendships have spawned from those in the years since.
Going to a place like that, taking courses to your heart’s content and hanging out with linguists for a month, is a sure fire way to fall in love with the field if you still feel like you’re on the outside looking in. I can’t recommend it enough.
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As far as PhD applications go, don’t worry too much. Just do what you think is interesting, put your heart into it, and send applications to places that you think will appreciate it, and appreciate you.