I've read some dialogues here and there but I want to explore

The Copleston histories are meant more for people who have been studying philosophy for a few years. There's a lot of untranslated Greek too.

There probably won't be a single right answer to this, but I'd read Copleston afterwards. He draws on all of Plato to basically give you the general themes of Plato's physics, metaphysics, ethics, etc., all at once rather than going dialogue by dialogue. So it's helpful to gather your thoughts once you've read those dialogues, but IMO is a bit premature to read on its own/before those dialogues.

Also Copleston is not at all difficult to read; I disagree with >>9939994 except to confirm that there is a fair bit of untranslated Greek in there. Read an intro like the Bloomsbury Companion to Plato before reading Copleston and you'll be sufficiently familiar with the Greek vocab necessary to have no trouble with Copleston.

>>9940136
I should have clarified that even though I'm no philosophy student I'm not a complete beginner. I've read introductions and works by other philosophers as well, including Kant and Hegel although not a complete work from any of these two.
I just want to start from the bottom and this time be systematic as I already said.
That's why I'm picking Copleston and not a history for dummies.
I have good Plato editions so there's quite a bit of intro there already, but what you said is what I wanted to know. How does Coplestone do it? Well then, it seems it'd be better to read it after the primary source.