The easiest way to invite friends, family or colleagues to join you on the Fediverse is to send them a link to JoinMastodon.org ⧉ or Fedi.Garden ⧉ or another Fediverse “on-boarding” website. Once they’ve signed up on a server, they can use their account to sign in on their server’s website or on a comptible app. For Mastodon, good third party apps include Tusky or Fedilab or Tooot for Android, and Toot! or Ice Cubes or Ivory for iPhone/iPad. There are also official apps for Mastodon, but they don’t have to use these, and the official apps tend to have fewer features than the third party apps.
Once your friends are on the Fediverse, you can exchange addresses with them and follow each other. You don’t need to be on the same server as them, following and interacting with each other is the same experience even if you’re on different servers.
However, if you do want people to sign up on the same server as you, there’s an “invite” feature on Mastodon which lets you invite them directly to your server. Log in on your server’s website, go to Preferences > Invite people, choose the settings you want for your invitation links and then click Generate.
Some servers close themselves to new sign-ups if they get too full, but still allow existing members to generate invitations. If your friend is unable to sign up on your server directly due to it closing to new members, they may still be able to sign up if you send them an invitation.