Hi Sara,
The meanings of 'fall over' and 'trip over' are about the same. The difference is grammatical. We can say 'I fell over' as a standalone phrasal verb. But 'trip over' isn't a phrasal verb... so if I say 'I tripped over', because I've included the preposition 'over', I need to include its object (the object of the preposition).
So, someone falls:
I don't know what happened; he fell over!
I don't know what happened; he fell over the edge of the cliff!
He tripped.
He tripped over that rock.
NOT: He tripped over the edge of the cliff (here you can see 'over' acting as a preposition and not as the particle in a phrasal verb... and you can't really trip over the edge of a cliff, which is why this sentence doesn't work).
AND NOT: He tripped over.
'Slip' is different and is usually used with 'on', as in 'He slipped on the ice'.
I hope this helps!