Lemmy doesn't really target compatibility with Mastodon. It does have some of it by using the same federation protocol, but it's all incidential and not actually directly supported.
If you wish for proper support, I recommend switching to Mbin instead. It's a Lemmy-like project that aims to work with both Lemmy and Mastodon.
When it comes to communicating between Lemmy and Mastodon though, this is what I know:
Contacting specific Mastodon users
You can mention any Mastodon user the same way you'd mention a Lemmy user. They will get your mention and will see the post or comment you mentioned them in. Your instance doesn't need to be federating with the Mastodon instance in question for this to work, as long as you're not explicitly defederated from each other.
Federation to Mastodon
Lemmy communities show up on Mastodon as users, so Mastodon users can browse and follow them. They basically function by boosting (retweeting) every post made to them. So all you need to do for your posts to show up on Mastodon is to have a user on there follow the community you're posting in.
Posting to Lemmy from Mastodon
Mastodon users can post to Lemmy communities by mentioning them, as if they were a user. Lemmy will display them as threads despite them being microblog posts, Mbin separates Lemmy-style threads and Mastodon-style microblog posts in your feed.
Discoverability
Interacting with Lemmy communities directly isn't too common for Mastodon users, hence the low amount of contact between the two. If you want to increase your discoverability, add hashtags to your posts. Mastodon iirc mainly relies on hashtags for discoverability.
Lemmy does NOT let you browse Mastodon posts or follow users on there. Mbin does though. So again, if this is something you want, do consider switching instead.