Is it okay to use the word "and" right after another sentence, and should I use a comma?
"And, as a member of my high school's entrepreneurship club, I participated..."
"And as a member of my high school's entrepreneurship club, I participated..."
Should there always be a comma after
andwhen it is used to start a sentence?
No. However, in the examples you give above, having the comma is correct. This is because the clause
as a member of my high school's entrepreneurship club is an appositive (an extra bit of information not necessarily needed to form a complete sentence). Appositives are typically surrounded by commas (but may also be surrounded by parentheses, like this phrase, or em dashes if they are fairly unrelated to the sentence's primary topic).
If you are using
and as an appositive itself (which is common when starting sentences with it, but not universal), then whichever stylebook you follow will have its own opinion. I know, for example, that The Economist's stylebook says that appositives at the beginning of sentences (if they are short) need not require a following comma.
Having said all the above, the more pressing issue is whether or not you should be starting a sentence with
and at all. Typically, starting a sentence with
and indicates that you can rework the previous sentence and add the following clauses to make the whole thought more succinct and well-worded.