Although verbs have often been called 'doing words' when the part of speech has been introduced to young children, this is a crude, partial, definition which can cause problems later on.
One set of verbs that need special explanation are link verbs, which show a property of (or redefine) the referent of a sentence's subject (though some may also indicate a transformation – from or to the said state, say):
Mr Cameron is the Prime Minister.
John is a vet.
Jill grew tired.
(Imagine a young child naively saying 'Mr Cameron - Prime Minister.' or 'John - vet.' to see the linking function that adult grammar demands.)
Another set is delexical verbs, where the verb is usually used to describe an action, but is really bleached of any particular meaning:
John had / took a bath.
Sally made a good suggestion.
My teacher gave / delivered a surprisingly interesting speech.
I'm including this rather long intro because one needs to consider what the terms 'idiomatic' and 'idiom' usually mean (and there is a large difference between the usual senses) before one can sensibly respond to OP's question.
(1) 'Give a speech' is a very common expression – it is 'idiomatic English.'
(2) 'Give' is not used in its primary (donate, cede etc) sense here; this is a delexical usage.
(3) We wouldn't say 'give a story' or 'give a tale', but we would say 'give a sermon' or 'give a lecture' – so the usage is rather idiosyncratic. This probably qualifies the expression as an 'idiom' in the usual sense.