This is the third time I've restarted this game and the first time this has happened. I'm doing the create this/improve this mini-quests and when I go to improve the iron dagger it says I don't have the required skill. I have basic smithing and can create the item. What is the required skill?

Is the dagger enchanted? If not, no clue what's going on. Iron daggers should be improvable by default.

Has anyone heard of this happening before? I am in the process of removing mods I had added before re-starting to see if any of them are the culprit but nothing's jumped out yet.

Okay, now I'm getting really frustrated. I've removed all the mods I'd added and I still get that message. So .... help? Please?

My smithing is completely broken. Can anyone help me figure out what's going on?

You aren't alone. I needed to raise my skill through the console REALLY high(4,000) to craft myself one of those 2500dmg daedric legendary swords..

player.setav smithing xxx(skill level here)

To restore it back to level you previously had, just use the command again and use the previous level value.

Smithing didn't function like this before...

If you are using any mods that alter the starting levels of skills you can see this happen. There is a minimum level for the Smithing skill in particular for you to increase an item to the next level beyond the default stats (Fine). The game expects you to have 15 as an absolute minimum so if you have less than this you will be unable to improve items.