If I wanted to say “I like speaking Irish” that would be “Is maith liom an Ghaeilge a labhairt”, right? Is that only for modal verbs, or does it happen in any sentence with two verbs, ie “I’m going over there and getting some food.”?
“Labhairt” there isn’t really syntactically a verb but a verbal noun (i.e., a noun based on the verb). The “a” that’s before it is historically a worn-down form of the preposition “do” (“do” and “dh’” are still found in this position in places in Munster, and “dh’” in places in Ulster, actually).
And there are lots of possible ways you could say your other sentence, e.g., “tá mé ag goil anonn go bhfaighe mé greim len ithe.”