Yes, you're quite right -- I got it upside down between reading and writing it out. The relative minor is a minor third below the major in question. So the relative major of A flat minor (G sharp minor) is C flat major (B major). Convention is, if your key signature has flats, stay with this when you modulate (likewise sharps), if at all possible.
And yes, you're also right that a shift from A flat major to C flat major (aka B major if you break convention) is a significant key shift, from the tonic to the flattened mediant: you'd write it as its IIIb (if I can use bs for flats). The mediant is a relatively distant key to modulate to, as 19th century harmony textbooks measured such things: the most distant being IV# = Vb, the sharpened subdominant or flattened dominant, which a six degrees away in a cycle of fifths: mediant and flattened mediant are both only four away, though in opposite directions. Anyway, four is enough for you to feel it, even in music that's less tightly harmonically bound than an 1820s textbook would accept.
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And yes, you're also right that a shift from A flat major to C flat major (aka B major if you break convention) is a significant key shift, from the tonic to the flattened mediant: you'd write it as its IIIb (if I can use bs for flats). The mediant is a relatively distant key to modulate to, as 19th century harmony textbooks measured such things: the most distant being IV# = Vb, the sharpened subdominant or flattened dominant, which a six degrees away in a cycle of fifths: mediant and flattened mediant are both only four away, though in opposite directions. Anyway, four is enough for you to feel it, even in music that's less tightly harmonically bound than an 1820s textbook would accept.