See also the official documentation:
Help Center > Members Manual > Joining groups

Whether you are new to Groups.io or you have already registered, the instructions are the same! You will automatically create an account for the email address you use. 

Joining a publicly listed group:

Go to https://groups.io/ and click on Find or Create a Group.

There is a list of publicly listed groups - https://groups.io/search

• You can search by keyword using the search box, by most popular (default), newest, and by name.
• Some will be restricted and you must apply to join them. Click on the link Apply for Membership in this Group
• Some are open to anyone who wishes to join. Click on Join This Group button.

If you have registered with Groups.io already with that same email address, that adds the new group's subscription to your existing account. If you visit your account's Identification page, you will find your profile in each group there.

Note: groups’ message archives may or may not be open to all. Check group policy in the list of groups. 

Joining a group not listed in the directory of groups:

Not all groups have chosen to be publicly listed. To join one of those, you will have get an invitation from the group or be given the subscribe email address.

If you have received an invitation by email: 

Remember that your email address is group ID. Give the group-owner your preferred email address to receive an invitation. 

Look for the invitation email in your inbox. Not there? Check your junk or spam mail.

Click and follow instructions.

If you have been given the group name or email address:

You can send a blank email to [GROUPNAME]+subscribe@groups.io  substituting the real group name for GROUPNAME.

If you used the same email address that you have used with Groups.io already,  your Display Name in the From header field will be copied into the new account's Profile, and from there into the subscription's profile of each group you join.

Whichever you do - join a publicly listed or a private group -  you will have a Groups.io Account. 

An Account is your email address, and is one way for you to control your Groups.io experience.

However many groups you join, there is no need to register your email address with Groups.io again.

You need not visit any Groups.io website if you just want to interact with email, 

but your Account is there and secure, even if you never set a Password (optional) with your Groups.io Account.

If you do visit any Groups.io website, you may need to Log In to access all of its features,

this requires your email address and either your Password or a Login Link.

see: Logging In and (optionally) How to Set a Personal Password

However you might want to customize your subscription for this group. See Help Center > Members Manual > Controlling your email subscription preferences.

Check out the rest of the help in this wiki if you wish to visit the group's site for message archive, or perhaps files, photos, etc. how to log in

You can also change your profile, etc. See Profile and Email Address and Merging Accounts