By default, new records are set to Only Me on the "Share with" setting:

To share the record with the rest of the team, select your Organization:

To share the record with all of your groups, click on the "Everyone" option up top:

If you only want to share a record with specific groups, click the box next to the group name:

Please note: While shared records can be unshared from groups, they cannot be reverted to private. To make a shared record private, you will need to create a new record with the same information, then delete the shared record.