I'm also new to this, but I think I understand the basics;

• I think beehaw does indeed run the same/custom software as Lemmy, and they are associated through the Fediverse.
• I signed up on beehaw, so assuming you can see this comment, you do not need to sign up on both
• beehaw is slightly different from other lemmy servers in that downvoting is disabled: I literally don't have the option to downvote you!
• servers can associate with any other servers they want, or block servers they don't like. For example, beehaw blocks lemmygrad (I think this one is quite commonly blocked, since its very large and visible, but also politically disagreeable to many users. I'm not sure if there's a way for users to filter out servers manually.)
• I think if a server is blocked by your server, you can still specifically choose to subscribe to it. I'm not entirely sure on this, however.

If any of what I've written is wrong, hopefully someone will correct me à la XKCD