Al principio

[fglorca]

Me voy a principios de junio.

Me voy al principio de mayo.

The second one is incorrect, am I right?

Al principio can't be followed by the preposition 'de', can it?
Al principio can only mean: in / at the beginning.

[Rusty]

Both can be used to mean nearly the same thing, in my view.
'A principios de' means 'around the beginning of', whereas 'al principio de' means 'at the beginning of' (note that the preposition is allowed). The former seems to encompass a few more days than the latter.