Where do I see my currently active spell or shout, that is, the spell I cast when I press Z? Is there a UI feature I can enable to see it?

it should be indicated in the magic menu, or the shorcut menu.

I meant within the game screen, not inside a menu

Not sure what command Z is bound to since I'm on Xbox and spells are set up L/R trigger and R bumper for shout/ability. But unless the UI is different, spells you get a hint from the hand shiny. Shouts you'll just have to remember. Its not that hard

The letter Z activates/launches the dragon shout on the PC.

OP, I have not seen anything on my screen when I launch a spell. Although on my mage, I can see the difference on her hands between a dual fire spell or dual chain lightning spell.

If its a disease top left.

If its a spell you have to check the magic menu.

Yeah, there is no icon like in oblivion. But your hands usually have a glow when a spell is active, and then there is a light effect when the spells wears off, so it is most of the time possible to keep track of things more or less. I do miss the icon in the top right corner from oblivion though.