Is it just as simple as googling it and if nothing comes up its ok, or is there more to it than that?
For the most part, but it’s still possible someone has a trademark on a name but it’s impossible to find it. Generally trademarks only apply if there’s a conflict of brand, but sometimes trademarks with legal teams can get a little overzealous. You can learn more about that here: https://en.wikipedia.org/wiki/Confusing_similarity
Once I’ve chosen a name do I need to copyright it?
Copyright is not for names and logos. Instead you should be thinking about getting a trademark. Generally speaking it’s not necessary to get a trademark, but you can do it if you feel like you need to legally protect your brand. If you ever end up working with a publisher, they may make you go through the trademark process. It’s possible to trademark your name after using it for a while.