Definitely get your wife's passport BEFORE the move. It will expedite things greatly.
You do need to be living continuously in Ireland in order to establish residency. Once you do that, then you can travel/work within the EU.
Establishing residency can be a hair ripping experience. The acceptable documents are never the ones you bring in, it seems. You must have a printed utility bill sent via An Post to you at your residence. A hard copy of your digital bill will not do, even if your utility posts it to you. It must be their PRINT bill, not a copy of an e-bill. The catch is that these bills are only issued every other month... Electric and phone bills are the gold standard.
Without establishing residency you cannot open a bank account or get a PPS card. The latter is required for nearly every activity you can think of. We did find that Ulster Bank in our area rejected EVERY document we brought in. PTSB Permanent, on the other hand, was much more lenient and got us set up with an account within an hour or so. So do shop around for banks.
If your wife and child are not Irish citizens they will need to register with the Garda. When we landed we had a lot of documents in hand to establish our intent to reside in Ireland. We were rushed through our entry and given non-renewable 90-day stamps on our US passports. This meant my husband needed to write to Dublin to the immigration people for permission to stay as the spouse of an Irish citizen before he could register with the Garda. We needed to send our wedding cert and other papers. The government staff are, by the way, very good about returning these original documents. Which is good, since copies won't do.
Before you leave you should accumulate some documentation of your intent to leave the US and reside in Ireland. We sold or donated tons of our personal belongings and kept all those receipts. We rented our house out so we had a year's lease on that. We sold two cars and brought the bills of sale for them.
Please do get that passport for your wife, if at all possible. My situation was complicated by the fact I've gone by my nickname on ALL US documentation since I was 14 years old. The Irish bureaucracy is pretty opaque and um, whimsical so the Boston Consulate advised us to wait to untangle that until we moved. I wish I'd gotten that taken care of years earlier and gotten that passport as our lives would have been much less stressful initially.
Having said all this, you *will* get through those first few months. No one's going to toss you or your wife and child out of the country. We fretted a lot about the expiration of my sweetie's 90-day visa stamp. I'm embarrassed to say it took quite a few weeks for us to realize that should the worse happen he would only need to go to Wales, not all the way back to the US, to reenter the country.
Our An Post postwoman, Bernie, was a joy during those first few months. She'd deliver an official message. We'd open it up on the footpath. She'd cheer or commiserate accordingly.
This is a great place to have your questions answered. The first few months are the hardest, but remember no one's going to toss you out. That was our biggest fear, but we were on less solid ground than you are. My citizenship was via Registry of Foreign Birth and not taken seriously until I got that passport. My husband is not a citizen. Even Irish who grew up here have a hard time upon returning.
Is it worth it? Yep, yep, yep!!!