As a dual national I would say definitely get both and keep both up to date.
It's true that a passport doesn't give you citizenship but it is a tangible official sign that you've gone through the process of proving that citizenship already (especially if it may involve providing parents/grandparents' birth certificates or other documentation) and you're not likely to have to go through that process again when you renew (unless you've left it for too many years, in which you might have to, which is why I'd say renewing is important too).
It also keeps all your options open. It might seem like wasted money right now but I would view it as an investment. You never know when systems might change, different rules or governments come in (I have been burnt by this in another area not to do with nationality). If you make sure your kids have both nationalities well documented (in the form of up to date passports) and usable (as a result of the up to date passports) then they will maximise the potential benefits from both.
Also, it can be handy to have two equally good forms of photo ID, in case you ever have to send one away and wait for it to come back, but that's a more minor advantage.
If I had to pick just one out of your two choices for now, I'd pick the Irish one to do first, assuming that requires providing documentary proof as it's a new passport, as I'd want to get that process out of the way and get that citizenship definitively established (even if it seems like it will be completely straightforward). Then I would renew the British ones as soon as I could and generally try to keep both up to date going forward.