Also, what's a typical path going from a team manager (EM1) to senior manager (EM2)? What are the things I should focus on to show I'm ready for managing managers? (I understand it can take a few years)
> I'm still trying to figure out a good model. Daily stand-ups?
One that I have used a few times with success was the right to left stand-up - instead of focusing on people, focus on what it takes to move a given task to the next column, starting from the rightmost tasks (finish work instead of starting it). This changes the stand-up from a status report to action planning. Some teams don't like that though, especially when people have some years of training in the standard yesterday I did X and today I'll do Y delivery, but I'll say worth a try depending on what you want from your stand-up once again.
Meeting with people, Hiring, setting up correct processes are your main tools.
(Qualifications: Director-level with several managers reporting to me in 60 eng org.)
You got a lot of answers for the first part, but not much for the second part. For the second part, I would recommend reading "The Manager's Path" by Camille Fournier. It's a great, short read that really lays out the different roles and responsibilities at different levels, and it does so at a high enough level that it should be relevant to most companies.
More concretely, this is a conversation you should have with your boss! The criteria for moving up in your career can differ from company to company, and your boss is going to be your best ally to get there. (If your boss is NOT interested in helping you, find a new boss!)
That said, don't skip too far ahead. Focus on doing a really good job with your own team, and once you've got that down and rock solid, then start to think about next steps of taking on multiple teams. It's kind of a totally different beast to manage multiple teams, but having the fundamentals down will make it more incremental than it would otherwise be. It's good to have a milestone down the road, but put 90% of your effort toward the short term and 10% toward making sure you're still moving toward those goals.
Good luck!
A lot of management is creating emotional safety so the team can self-organize and give/receive feedback safely, without fear of blame or repercussions. If the team owns the rituals, they'll be more committed to making the process work best for their needs. Of course, you have input, and even the final say if nothing presents itself or there's issues that require decisiveness, but you have to balance your top-down authority with buy-in from the whole team.
Usually these are things that are specific to a company culture, if everyone there does daily stand-ups do daily stand-ups. Same for the focus, try to get a feel from your boss about what the company is focusing on and try to emphasize that at the meetings. In general, if you don't have a passion or good reason for believing in a different method, it is better to follow the practices they are most used to. Make friends with someone else at your level and ask their advice.
> What are the things I should focus on to show I'm ready for managing managers?
At some level it stops being about the managing(who is working on what) and more about trying to give people the resources to maximize their value. Managing managers isn't really considered by most to be managing like it is for low level workers. It is about giving people the opportunity to improve the company and setting direction. It is also about saying no to the right things.
I'd also encourage you to bring this up with your team in retros - "Hey, I'm thinking we could try XYZ, what do you think?" or "I'm seeing XYZ as an issue, what's a better way to do it?".
For the team it gives them the feeling they can talk to the Manager honestly, which is really important.
As for career path, read The Manager's Path.
Has been really helpful. Footnotes are good.