This site is currently struggling to handle the amount of new users. I have already upgraded the server, but it will go down regardless if half of Reddit tries to join.
However Lemmy is federated software, meaning you can interact seamlessly with communities on other instances like beehaw.org or lemmy.one. The documentation explains in more detail how this works. Use the instance list to find one where you can register. Then use the Community Browser to find interesting communities. Paste the community url into the search field to follow it.
You can help other Reddit refugees by inviting them to the same Lemmy instance where you joined. This way we can spread the load across many different servers. And users with similar interests will end up together on the same instances. Others on the same instance can also automatically see posts from all the communities that you follow.
Edit: If you moderate a large subreddit, do not link your users directly to lemmy.ml in your announcements. That way the server will only go down sooner.
This is inevitable if feddit is going to become mainstream. People have a herd mentality, if Lemmy is going to become popular there will always be a handful of instances that are much more popular than the others. These popular instances will need to scale (both vertically and horizontally) while the smaller instances will probably keep getting by with a single server. This is the same way email providers work, half the people I know use gmail, and most of the others use another large provider like yahoo or hotmail. It's just the way this is going to have to work. People want to join an instance with their friends, even if they're all federated together. They want to know that the instance they sign up for has peer approval and it's already a tried and trusted one.