Joined a new company earlier this month, should I set up a skip level meeting?

You don’t need to tell your manager unless you want.

Also, I think it can be beneficial to you if you are senior/staff to get insights into what is the vision of the org and find opportunities and asking them how you can help.

But, if you have nothing concrete to ask or discuss, just keep it monthly rather than weekly or bi-weekly. I have it setup monthly and I cancel if I have nothing on agenda to save time for both

At Big Tech, it's not super common for senior engineers to have regular meetings with their skip, that's more of a staff-level thing. Regardless, you can try as it certainly wouldn't hurt to talk to them and establish yourself as a friendly face. Here's a good video to prepare for it: How To Effectively Leverage Skip Level 1:1s

2. Should I tell my manager I'm talking to my skip level ie his manager?

It's not a requirement to tell your manager, but you might as well since your manager would be the most effective "broker" to help you set up that meeting. Just tell your manager that you're interested in talking to the skip and see if they can help. If you have already started off on a good foot with your manager, they can put in a good word for you with the skip to make the meeting more productive.

On top of all this, I highly recommend the onboarding course: The Complete Onboarding Guide For Software Engineers