Neither of those are common-use pronouns, but for different reasons - one isn't common-use, the other isn't a pronoun. I'll explain.
我が輩 is a relatively unusual first-person pronoun. It is used in exactly two contexts:
- When a male speaker wants to sound stuck-up and self-important - almost always in fiction, and often with noticeably more literary speech than is normal for conversations
- When someone is making a reference to 「我が輩は猫である」, the first line of a very famous Natsume Souseki short story (though the line is arguably more famous than the story as a whole)
No speaker outside of fiction would ever unironically use 我が輩, though it may show up in jokes.
As for 方, it's not a pronoun, but rather a pluraliser suffix (read がた thanks to rendaku). It is, as you say, a more formal version of 達, and it's used primarily to show deference to the people whose noun it's attached to. It's most commonly heard in set phrases like 先生方, but it can be attached to anything as the situation demands. Still, outside of set phrases, you're most likely to hear 方 used in situations where the people whose noun it's attached to are of high enough status to require verbs like なさる - it's not too much of an everyday word.