Is there a legit way to read scripts for movies and/or TV series? I tried to google it and found some sites but they all seem... quite fishy. So I'm asking here.

Try the BBC writer's room:
http://www.bbc.co.uk/writersroom/scripts

It's got sample scripts from various BBC shows and loads of tips etc for writing scripts (or it did last time I really looked at it, which was a few years ago)

Now this looks decent! Thank you.

No problem, hope it helps :)

Are you looking to read specific scripts or are you looking to have examples to go by? Celtx is a really good script-writing program.

Examples, mostly. Guides how to write them, how much info to include per scene and stuff like that.

Celtx has it all formatted and spaces to make notations and kind of guides-as-you-write if you're not familiar.

IMO (as a want-to-be/knows quite a bit about it all director) the more information you provide with your writing, the better, so the director can really understand what you're going for, but, of course always try to express what you're going for as succinctly as possible (hahaha *laughs at self*)

The Hollywood-ized industry has screen-writers paring down content to a detrimental level. Write what you want to be sure gets depicted.