How do you get your Champions to learn new combat spells?

Have you imbued your champion? That enables them to cast spells. Each time you select a spelltrait when a champion levels up, you should gain access to new spells. Some of these are strategic, some are tactical.

Cast "Imbue champion".

Without that, a champion can't use magic.

Also, you can find or buy scrolls and use them to learn new spells. There is a special building that allows this, 

Yep, I imbued them all, but still no tactical spells, the only tactical spells I have come from an Item or the one healing spell I have from the Level up Perk "Healer",

Other than that my mages are strictly melee fighters  :(   So of 5 champions I have one that can cast heal, and one that can cast blind.  Not very much fun for the mages :'(

When they level up you need to pick up the various magic levels for them (i.e. earth apprentice, death disciple, etc.).  This opens up more spells for them to use.  I.e. if you want to use burning hands you have to be able to use fire magic.  Hope that makes sense. 

Of course you have to have found that spell or researched it somehow as well.

Can you tell us specifically what spell traits you have unlocked?

For example. my Sovreign has Life Disciple and can cast heal as her only tactical spell in combat

Bacco has Earth Disciple with no tactical spells...

All of my Champions have access to the strategic spells that they can cast outside of combat but unless they have an item equipped that gives them some tactical spell/function my options in battle are limited to melee attacks or item use.

Perhaps Im not researching spells properly, Ive researched some of the "books" but perhaps Im not focused on the right things or I havent built the buildings.

Did you play the tutorial?

The main way to get spells is to select those traits when your champions level up. Life and Earth in particular just happen to unlock strategic spells in the start mostly.

Yep Played the tutorial and it didn't really get into advancing your magic  I do like the tutorials and hope they get expanded.

For grins I started a new game, this time when I hired Champions, they had the tactical spells and when leveled up new ones showed up as well.

Seems like my first game is hosed in a number of ways.  I also lost the ability to have my explorers explore automatically and some buildings in my "Build List" could not be selected for building.....  Thanks for the help I think what I have here is a corrupted save game. I just kept resaving over the same slot.

Same thing happened to me, but it seems gone when reloaded the day after...