I'm starting to get my first surfboard lessons and wanted to get my own.
Should I start off with a soft board or a hardboard?
I recently went through this decision myself. Something I was told is you should be able to consistently stand up/catch waves on your current board before you move to a "harder" board. So by this advice you start on a foamie. I went for a cheap ($50) slightly used foamie. Of my 5 friends who also just started we split 2/3 foamie/fiber. I think the 3 that went for a fiberglass board (used ~$100-250) are happier. I'll be buying a fiber board soon. Now that that anecdote is out of the way onto advice.
Things I was told to consider:
A lot of places(/friends) let you demo used fiber boards for cheap. Some will even put the rental cost to buying a board. If possible, try demoing one and if you are able to get up consistently then just buy a fiberglass board.
Generally it's recommended to get a "foamie" as your first board, mostly because they're relatively cheap so if you don't commit to the sport you're not so much out of pocket on the attempt.
By the time you're looking at upgrading you should know a lot more about the type of surfing you want to do and what kit you're going to need.
Should I start off with a soft board or a hardboard?
Foam board for sure. Primarily because an early bad experience (e.g. injury) could deter a learner from continuing to try, and if you get hit by a traditional board you will feel it. They are also really fun to ride, they float well, they paddle nicely, and they don't get dinged. Spend a summer on a foam board until your soul has changed permanently, then invest in a used traditional glassed board.