Sabaton is great, I started listening to them years before I really got into metal. All those bands you listed are considered "power metal". I think it might be the best sub-genre to begin acclimating with, because it's so melodic and there aren't a lot of harsh sounds.

You might also like doom metal, which I'd consider to be a bit heavier (thought the bands vary greatly), but still primarily focused on melody, with lots of clean vocals. Here are some bands I like, and I'll link to one song on YouTube for each:

• The Sword: probably the best one to start with if you're coming from power metal, they have the same high energy and corny (not an insult) vocals
• Black Sabbath: widely considered to have invented the genre. All metal can trace it's roots back to them but I'd say doom metal emulates them the most closely. They're a product of psychedelic rock and disillusionment with the flower child movement, and they wear a lot of blues influence on their sleeves.
• Sleep: sort of like what Black Sabbath would sound like if they smoked 10x as much weed, or power metal if it was slowed and dropped an octave. They're the biggest departure on this list from what you listen to, but they're also my favorite to share because they're the band that really got me into metal.

My wife, who is not a metal head, was amazed to discover the vast number of sub-genres within the broader metal community. Like what you like, other people can like what they like and that's ok!

As a classically trained musician, have you looked into Tool at all? Their song construction is amazingly complex and fascinating.