What harm can 9V do to common electrical components
It depends where that 9V is coming from. If it is a series of 3 rechargeable AA-size or similar lithium cells, you get a fire if there's no fuse/PTC to limit the current. A hookup wire used for connecting things in the kit will happily glow orange when supplied from such a battery. Transistors and diodes connected randomly to such a source will crack open, flash, and release magic smoke. Etc.
If the 9V is coming from a small zinc or alkaline battery, then it can damage or overstress components, but the damage should be contained and not particularly hard to get rid of by replacing the components.
I am considering having everything attached to a common ground by default
That will make the thing useless. Look at just about any schematic of a circuit for such kits, and tell us how many components are not connected directly to ground.
The kit you show has nothing connected to the battery by default at all, in any way, and that's how it should be.
To prevent something bad from happening.
Grounding things only prevents some very specific bad things from happening. None of that applies to this kit, more-or-less.
On the contrary, that will make bad things happen by default - especially shorts between +9V and 0V. With alkaline batteries, the battery will just be discharged quickly. With rechargeables, things may actually catch on fire if you're unlucky. Do not use rechargeable batteries in those kits. Zinc or alkaline are fine.
What could possibly go wrong if you connect any of those components wrongly.
Nothing wrong at all - other than the circuit not working - to a component getting very hot and releasing the magic smoke, then going open-circuit or short-circuit or something between the two.
Resistors typically break open as they overheat and the conductive material vaporizes and/or catches fire for a short moment.
Transistors tend to blow their plastic package open when overstressed with vigor, but can also go short or have degraded performance from non-terminal overstress.
The meter should be internally protected against overvoltage and overcurrent in a kit like this. When adding your own meter, you'll have to design the protection mechanism. It can be very rudimentary, say antiparallel diodes connected to the movement if it is an ammeter. If it's a voltmeter, just make sure it can take 9V, and it should be fine. For higher-sensitivity voltmeters, a voltage clamp circuit will be needed, and perhaps a series PTC thermistor to reduce current flow when overloaded.
Could it start a fire with a 9v battery?
Yes, of course, but not very likely by accident. It would take some design work to get something that could get hot enough to start a fire without destroying itself first in this kit.
Could some components break?
Always. That's part of learning. I'm not sure where the idea that you don't break any components while learning, or that breaking them is somehow very bad, came from. Real life doesn't work that way. If you don't occasionally break a component, you're not trying very hard :)
Most components in this kit cost on the order of 1USD. Breaking them while learning is very cheap compared to the value of education one gets from doing it.