Can magical spells actually damage objects?
So something standalone, e.g. the pile of wood over there in the corner of the room?
And on a character, e.g. the quarterstaff in a character hand?
[I think they can, but I'm concerned I'm confusing DnD 5e with earlier versions]
The DMG (pages 246-247) has specific information about how objects can be damaged including ACs for different substances and suggested hit points based on size and resilience. These rules can easily be applied to damaging spells too.
I'm not going to reproduce the whole entry, but your pile of wood would probably have an AC of about 15 and hit points of about 3d8 (if it's loose branches) or 4d8 (if its cut lumber).
Please bear in mind that many spells specifically state that the target is a creature. I would probably disallow such spells from being used against objects (unless they've been animated and are attacking the party!).
There are a couple of entries which are quite important though.