The most important thing, other than experience and exposure to a field, that students receive from an internship is the connections you have made with colleagues. When you finish your internship, I personally feel that thank you notes (or emails if your handwriting as bad as mine) are a great way to leave on a good note. There’s a chance that when your colleagues think about you and your performance, they could think of the thank you note, so they’re worth putting a good amount of time and thinking into.
The tone of the thank you note can vary based on your relationship with the colleague you’re writing it to. However, it should remain professional, so having only inside jokes is probably not the best idea. When I wrote my thank you notes, based on the fact that the office was small and I knew everyone, I tried to refer to learning opportunities I had from the summer involving the colleague I was writing to. For example, I could write to someone thanking them for showing me how to use software during my first week.
In terms of who you decide to write thank you notes to, that’s up to you, but my philosophy on it was that it’s better to write too many thank you notes than to write too few. If people, especially in a small office like I worked in, are talking about your note on the week after you leave, it allows for the opportunity for people who didn’t receive them to feel left out and unappreciated. Since I appreciated and genuinely enjoyed connecting and learning from all of my colleagues, I wrote them each an individualized thank you note.