If I have a British passport and birth certificate, do I need a green card?

It depends. If you plan on living in the United States, then you will need to become a lawful permanent resident. As a result, you will definitely need a green card. A British subject is allowed to apply for a visitor visa. This can be done online without a visit to the US Embassy. However, you must intend to visit for less than 90 days. There are some disqualification. You may want to consult with a competent and experienced immigration attorney before you file for a non-immigrant visitor admission. Good luck.

The answer to your question depends upon what your immigration-related goals may be. If you may wish to be permanently lawfully present in the U.S. (beyond the period authorized by treaty or a visitor's visa), if you may wish to become legally authorized to be employed in the U.S., if you may wish to be able to sponsor certain family members for immigration benefits, if you may wish to become on a pathway to eligibility to become a naturalized U.S. citizen, etc., then applying to become a Lawful Permanent Resident (getting a "Green Card") may be important to you. It would be wise for you to consult with an immigration attorney about immigration-related eligibilities, options and strategies to attain your goals.