What is a good place to find science-related news, without having to pay a subscription?
Personalise your home page, you can add topics there, and you get automatic updates.
You can also use sites such as this (National Geographic) for free.
How Would I Go About Becoming a Freelance Writer For Science News?
Simple, really. Find the name of the chief editor or managing editor of Science News and write him/her a letter telling him you would like to become a freelancer and explain that you have a passion for science writing.. You should include samples of any articles you have written and a resume, showing your academic and work background. Hopefully, they have some science connection. Good luck!