Data Consistency Between Two MySQL Databases Running Different Versions
Plan A:  Do it one step at a time.

Plan B:  Set up a long replication chain:  5.5 -> 5.6 -> 5.7

Plan C:  Take some downtime to do it via a dump.

Plan D:  "Update in place".  That is, install the next version on the existing version, keeping the data intact.  Be sure to run mysql_upgrade.  Repeat.

Caution:  The upgrade from 5.7 to 8.0 may be the trickiest.

Caution:  If you are using PHP's mysql_* interface, it would probably be better to switch to mysqli_* first.  mysql_* has been removed from php 7.


  I also want changes to the database on the new server to reflect on the database on the old server


Abandon that goal.  There are too many risks in using Dual Master with different versions -- primarily because the "Slave" needs to be no older than the "Master".

Get a separate machine to practice on.