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Short answer: they are the same language.
'Persian' is the name of the language in English, and 'Farsi' (فارسی) is what Persian speakers call it themselves — just like 'German' in English is 'Deutsch' in German. When you search for Farsi lessons or Persian lessons, you are looking for the same thing.
There are three main varieties: Farsi in Iran, Dari in Afghanistan, and Tajik in Tajikistan. They share the same roots and are largely mutually intelligible in speech; the biggest differences are accent, some everyday vocabulary, and the writing system (Tajik uses Cyrillic).
So which should you learn? If you are interested in Iran, Iranian culture, travel, family, or tests like the DLPT and OPI, learn Iranian Farsi — it is the variety I teach. My lessons cover speaking, listening, reading and writing with the Persian alphabet, step by step.
New to the language? Start with my beginner lessons page: https://www.shadi-davarzani.com/learn-persian-farsi-online-for-beginners-lessons-with-a-native-tutor
Book & Pay for Lessons — One term is 4 one-hour sessions for $80, paid in advance via Venmo. Your day and time are reserved once payment and the signed agreement are received.