'Ba mhaith liom labhairt le Máire' = 'I would like to speak to Máire'.

I have heard somewhere, can't remember where, so don't know if it is reliable, that 'caint' is talk and 'labhairt' is speak. They seem to be interchangeable in English or perhaps misused? Anyway, bottom line is I am confident my use of 'labhairt' above is correct, but am unsure about when to use 'caint'.

(By the way, 'saoirse labhartha' means 'freedom of speech'!)

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Nothing wrong with
Ba mhaith liom caint le Máire
either...
... or
saoirse cainte

... but
Gaeilge labhartha = spoken Irish
an teanga a labhairt = to speak the language

... but pretty much interchangeable in many cases I would say...

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Bí cinnte de go nglacfaidh triúr le gach aistriúchán a thabharfar.
Be sure to get three in agreement with a translation given.

For some reason my teacher always discourages the use of "caint". She corrected me many times to use "labhair" so that's about all I use now unless I want to say "chat". Maybe it's an Ulster thing as her people and her Irish are from Toraigh Island. Lughaidh could comment.

Labhair mé le Máire.
Bhí mé ag caint le Máire.
Ná bí ag cúl-chaint.

Bhí mé ag caint Béarla leis - sounds odd.
bhí mé ag labhairt Béarla leis.

I don't know what the grammar rules are. You just know when to say which one.
Like Braoin said they are interchangeable.

How about these:

Bhí mé ag caint as Béarla leis
Bhí mé ag labhairt as Béarla leis

I'm not sure why, but "caint as" seems more natural than "labhairt as."

That is probably more Donegal or maybe Munster.

I would never say "as Gaeilge / as Béarla"
I'd say - i mBéarla / i nGaeilge

I think it's Munster, but it's also the form taught in Galltacht schools, because everything is simpler without the pesky urú.

Who actually says, 'as Gaeilge' in their dialect then? That is what I always said and have only recently started saying, 'i nGaeilge'. Of course, my Irish is school Irish.....

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It's that dreaded Caighdeán then