Hi Scott and welcome to caving!
I've never done a lot in North Wales and I have heard that the politics can be problematic in that area. There are several forum users in the area though, who I'm sure will be along with some advice.
Every area I have caved, local clubs often offer beginner/ taster trips and are always keen to recruit keen new members, so I hope what you've heard about it being difficult to join a club in North Wales isn't actually the case.
Usually, you would email the club and they'll tell you when the beginner trips are, or otherwise organise something specifically for you. Once you've been on a few trips and got to know a few people, you'll be able to apply to join. Obviously clubs can't hand out membership to people that nobody knows as they typically loan out equipment, so they do need to know you're not going to kill yourself with it
This isn't about keeping people out, it's about keeping people safe.
With coronavirus, things are a little different at the moment. Different clubs are handling things in different ways depending on their membership and facilities, so you might find it's a bit less straight forward than usual...