Lots of useful info here, well done everybody!
I would just add:
1. I've done "family reunification" and it's pretty straightforward.
2. You mention "partner" while others mention "wife". Civil partnerships and "durable" (unmarried) partnerships (more than 2 years) are recognized in the EU, but I found Spain to be quite vexing about the proof for said partnership. A marriage certificate for a wife is a lot less hassle!
3. "Residency rights" come with residence, which is pretty much immediate with your first "temporary" (5 years for EU citiziens and their partners) permit. As a resident there should be no discrimination between you and citizens (this is part of EU directives of free movement). However, 5 years is the timing for switching to "permanent" residence. It should be noted that permanent residence, granted in any EU country, is pretty much equivalent to a passport (aside from right to vote), and allows you to live, work, study and retire in ANY EU country, on the same basis as EU passport holders (ditto EU free movement). The permanent residence permit also has protections in terms of indefinite renewals and very restricted grounds for rescinding it. So while one might feel slighly more emotional about having a passport, the permanent residence permit is very nearly as good in practical terms.
4. There is nothing to stop you pursuing your dream of being retired in Spain, by obtaining legal residence in both Spain and Ireland. I hold residence permits in Bulgaria, Cyprus and Spain with property in each country. The initial temporary residence you can hold in many countries with no issue. However, you can only hold permanent residence in one EU country (hence why it grants pan-european rights). So you can find your retirement home, and enjoy both countries, while accumulating time physically in the country and being legally resident.
5. It's probably slightly easier to have a simple progression in one country from temporary, to permanent, to citizenship. But I don't think it's strictly necessary. You could have permanent residence in Ireland, but end up applying for Spanish citizenship, or vice versa. It just depends on the rules for citizenship, which are typically to do with time in the country. Spain does have a language requirement, but someone who already speaks 2 languages is unlikely to find this a barrier.
So for me, I'm currently spending more time in Bulgaria (because of the Covid Crisis), so I failed to get over to Cyprus and claim my permanent residence during the "transition year". But I didn't panic as I'll get permanent residence in Bulgaria next year, instead. However, it's unlikely I will apply for or get a Bulgarian passport as I am not learning Bulgarian. Whereas my Spanish is already above the level needed, and my partner will soon be a Spanish citizen. Likewise, I'd expect an Irish passport to be easier / quicker to get for the partner of an Irish citizen. By way of contrast, my partner is spending more time in Spain, and she's skipping permanent residence, and applying directly for her passport this year (Ibero-American exception).