That's a tough question!
The meanings of these three verbs are all very similar, but there are a couple of differences:
'to bloom' is usually only used for trees/flowers, and most commonly the phrase 'in bloom/blooming' is used (as a state):
e.g. Hopefully the roses will bloom in May.
e.g. The roses are in bloom, and they look lovely! OR The roses are blooming.
'to flower' can be used for flowers:
e.g. The roses are beginning to flower.
e.g. The roses are flowering.
'to blossom' can be used in exactly the same way as 'bloom' and 'in bloom/blooming' ('in blossom/blossoming') - but usually more about trees than flowers.
e.g. The cherry trees are in blossom OR The cherry trees are blossoming
It can also sometimes be used to describe people/things:
e.g. Jane has been working hard at school and she is improving, she is really blossoming into an intelligent young lady.
Hope that helps!
If that was confusing or you have more questions, feel free to ask me,
Jonny