Because your wife is an Irish citizen, you are the spouse of an EU citizen. You will therefore continue to have rights under the free movement directive even after most citizens of the UK cease having those rights. In particular, as long as you travel with your wife (or to join her), you are a beneficiary of the directive. So in fact, very little will change for you as far as traveling with your wife is concerned. This would not be true if your wife were not an Irish citizen.
Because you qualify for free movement, you may remain indefinitely in any EU country provided that you have sufficient resources (or your wife is working, studying, or looking for work, but these don’t seem to apply to you). You may spend up to three months in each EU country without having to register. You may be required to register with the authorities to remain longer than three months. The details of this registration depend on the country in question — some don’t require it at all.
The 90/180 rule that applies to most short-term visitors in the Schengen area does not apply to you. This is explicit in the Schengen Borders Code, which excludes beneficiaries of the directive from the provisions applying to nationals of so-called “third countries.”
Would I need a visa to do this after Brexit for each country we visit, or can I only stay for 90 days?
The UK is going to be on the visa-exempt list for the Schengen area (otherwise known as “Annex II”). Because of that, family members with a derivative right of free movement may only be required to have a visa if their country of nationality is an “Annex I” country. In other words, you will continue to be able to visit the Schengen area by yourself for short visits without a visa, and you will not need a visa in the Schengen area when you are there with your wife.
(If the relationship between the UK and the EU should become so strained that they introduce visa requirements, then you would also need a visa to travel with your wife, at least in theory, but family members crossing at the land border are generally admitted without such a visa provided they have satisfactory evidence of their family relationship. Furthermore, the possibility of such a severe deterioration between the UK and the EU seems very remote.)