"Labhraím Gaeilge."

Lol, same here.

The way you were taught was correct. Ability to speak a language is expressed by "Tá _ agam".

Yeah, like @smrch said, "Tá _ agam" stands for "I can speak " in this case, "Labhraím Gaeilge" lacks something, like "Labhraím Gaeilge ar maidin" (I speak irish in the morning)... but i'm not sure, i think it goes like that, like actually SAYING something

Gaelach

Indeed. And there is a difference between Tá Gaeilge agam and Labhraím Gaeilge. The former means ability, whereas the latter means that you habitually speak Irish.

No, that wouldn't be correct as the term, 'Labhraim Gaeilge is incorrect.' You could say, 'Ta Rúisís Bealarúisís agus Úcráinis agam.' In Irish we talk about HAVING a language. When we say we SPEAK a language we're expressing that we speak it daily, or on certain occasions, etc.

Here you go (click on "Present" in the drop-down menu).

I want to really have the ability to do it! =)

Because Irish is never called Gaelic. Only Scottish Gaelic is referred to as such to differentiate it from the Scots language. Irish is always called Irish in English.

Technically, you can in the US and Canada, as the Irish who came here called it that as time went on (Irish teachers from Ireland have also told me it's not incorrect here), and we have a rich tradition of calling it that throughout North America, but in Ireland when they say Gaelic they're either referring to Irish football or the Gaelic language of Scotland. Also, since there are three Gaelic languages: Gaeilge, Gàidhlig, and Gaelg, you're better off just calling it Irish, Irish Gaelic, or simply Gaeilge (the actual name).

Well, "Gaelic" is the way it's called in English, but in Irish Gaelic it's called "Gaeilge".

It's just called Irish. On occasion you may here someone call it Irish gaelic to differentiate it from scottish gaelic(A closely related language) but besides that it's just Irish.