Citizenship through parents -- apply for citizenship formally first or go straight to a passport?

Hi! My siblings and I are born in the US. We'd like to work on getting our British citizenship established and passports. Both of our parents are born in the UK. Per the immigration website, we're eligible for citizenship through them. Is that citizenship automatic or do we need to formally apply? Would we need to have the citizenship clarified (or paperwork confirming our status) *before* applying for our passports? I spoke with the UK consulate in my state last week, and the representative was just as confused as me. Thanks in advance. :)