visa, citizenship, teaching English

The question which you should be asking is whether or not you can actually work in Europe?

As you are from the USA you may find this task to be more difficult than you may imagine.

This may help Eire

It's pretty difficult for an American to get a work visa there. You'll have better luck elsewhere. You might want to do a few searches online.

Sorry to burst your bubble but unless you qualify for the scheme in #1's link you have absolutely no chance of being allowed to work in Ireland at the moment. Work permits are only granted for very specialised skills where there is a shortage of qualified EU citizens. English teachers don't qualify, even trained ones.

Your only chance of working in Ireland is with working holiday visa. You need to be 30 or under when you apply, and is valid for a year. As an English-speaking country, I don't know what the job prospect is for an American, even with CELTA qualification. Clearly you can't teach kids in normal schools, as they teach to an advanced standard and you will need teaching qualification. This only leaves language schools, for those coming over to Ireland to study English, but I'd have thought most schools prefer speakers of British English, rather than American.
Citizenship? Remember, as an American, if you voluntarily apply for another nationality (and not born with it as a dual national), you risk losing your US citizenship. This is a grey area, with lots of exceptions, so you need to investigate the rules as they may apply to you. In any case, you need to be living legally in a European country continuously for 5 to 10 years before you can apply, and there're often stringent requirements about how well integrated you are in the local community, language fluency only being one of them. Almost all countries guard their nationality jealously, and don't give it out readily, and only to those who will continue to make useful contributions to their adopted country. Often the easiest is marriage to a national, followed by several years' residence.

There is actually virtually no chance of losing your US citizenship merely by taking out a second citizenship. The State Department will presume you intend to keep your US citizenship absent extraordinary circumstances (such as joining the army of a country at war with the US). So you don't need to worry about that.

I don't know about the citizenship requirements of other countries but #4's advice in that regard is probably generally sound. Ireland doesn't (at present) have any requirements in relation to integration etc, but it does require five years' legal residence and the application procedure itself takes a few years.

#5
Some countries, Germany for instance, make you renounce existing citizenship before allowing you to be naturalised. While it's a moot point whether such renunciation carries official meaning in the eyes of US State Department, it's not something you should enter into lightly. Britain and Ireland have liberal policies on dual or multiple citizenship, but they are an exception rather than a rule in the modern world.

Germany can refuse to give you its citizenship if you won't renounce your US citizenship but it has no power to deprive you of your US citizenship upon naturalisation as a German citizen. That's a matter for US law alone. Obviously OP needs to know all the rules of whatever country s/he seeks naturalisation in, but s/he doesn't need to fear that merely by taking out (or even applying for, as #4 says) another citizenship s/he will risk losing US citizenship.

BTW unless things have changed in the past few years the US is one of those countries that officially requires you to renounce other citizenships upon acquiring its citizenship - but that's a mere formulation of words with no binding legal effect.

#7
I'm not so confident of 'no consequences' for US citizens taking out another nationality voluntarily. There have been cases of people receiving sanctions for breaking rules. While it's unlikely to happen in case of citizenship of a European country, a change in political situation, and its relation to US, can have an adverse effect on dual or multiple nationals.

Perhaps you could provide some examples of such consequences #7 (real life consequences I mean, not hypothetical). The State Department's own web page acknowledges what I said in #5.