Graduated with Major in Computing Science and Business 9 months ago.
Eight months of intern development experience at a large (50k+ employee) company, doing front end web development.
Further eight months as QA/Automation tester intern.

Currently unemployed.
No side projects.
I have almost no experience with system design, backend, scalability. Not sure how to go about learning these concepts.

Goal is a job at Amazon or Microsoft (the only big tech employers in my area).
What should I be focusing on to get there, and in what order:

Grinding problems?
Working on a side project to gain more experience on backend?
How many months should I be allocating to prep?
How do I balance between getting better as a developer vs getting better at interviews?

Money isn't (currently) an issue but it's difficult shutting off the voices in your head that you're the disappointment of the family.