How to keep the Kings from fighting each other

Succession laws for multiple kingdoms (and no emperor title)
(For your example where you are the King of Ireland and Wales)

1. King of Ireland and your lower titles that are de jure part of Ireland are governed by the succession laws for "King of Ireland".
2. King of Wales and your lower titles that are de jure part of Wales are governed by the succession laws for "King of Wales".
3. [Duke and lower] titles that are not de jure part of any of your King titles are governed by the succession laws for your primary title

Changing a succession law
If you changed the succession law for King of Ireland, and King of Ireland is your primary title, all of your de jure Irish titles and titles outside any of your de jure kingdoms (#1 and #3 above) will be governed by the new succession law.

You would need to change King of Wales separately. (In the law interface, click on the coat of arms for Wales.)

Succession laws for titles held by vassals
Your vassals have their own separate laws for succession which could be totally different from yours. For example, you have have agnatic primogeniture as your kingdom succession law, and your vassal duke could have agnatic-cognatic gavelkind as his succession law.

Finding out why they hate you

To find out why people don't like you, hover over the opinion number to find out why: (alas, the cursor isn't showing in this screenshot. I assure you, it's hovering over the number.)

You can find out why people don't like you from any screen that shows their opinion number. Here's another example:

Doing something about them hating you

Vassals often rebel because they don't like you. Once you've worked out why they hate you, try to do something about it. If you have 98% of the titles in a realm, then, as you mentioned in the comments, they probably think your demesne is too large. (Fun fact: "demesne" is actually pronounced like "domain." Language is weird.)

To fix both of your problems (demesne too large, nobles hate you when you give out a title), you need to give out a lot of titles at the same time. Your demesne limit is in the upper-right corner of the screen; for you, it probably says something like "20/5", which means your demesne limit is 5, and you have 20 titles. The "20" will be red, and there's probably a notification at the top of the screen talking about that (unless you made it go away).

You need to do three things:

1. Pause the game
2. Give away all your excess titles, until you're within your demesne limit.
3. Give away some excess duchies, until you're down to 2. (You get -10 opinion against all of your vassals per duchy above 2.)

If you can, arrange it so that you have 2 duchies, and all of the counties within those 2 duchies, and you're within your demesne limit. Spread out the titles a bit, so you don't have one Super Duke who's going to challenge your power.

There's some other advice for preventing revolts over at this other Arquade answer, but this will get you started.

Many thanks to @svick for figuring out the actual problem here.

(As a side note, you probably also have some note about being the "wrong holder type" for your titles. You're a noble, you'll get the best benefit from holding castles (i.e., baron titles) and counties (count/earl titles). Give away your cities and churches to other people. (And when you hit your demesne limit, give away the baron titles and keep the count titles.))