When should I apply to companies?

I read or saw a video a long time ago that when you negotiate, you want to have competing offers so that you can stack them against each other.

This will always be true, but I think this advice is less applicable in the current market. Of course, if you can to pull this off, that's great, but I wouldn't stress out too much if you aren't able to get these stars to align. Just having a decent job in this economy is a win.

Generally speaking, what is the correct timeline for applying to big, medium, and small sized companies?

I would say it's more about your excitement level for the companies rather than the size.

You will get better at interviewing over time as you shake off the rust in the beginning and then learn from your mistakes. You don't want to do this with companies at the top of your excitement list as, well, failing their interviews would suck.

Start off with the companies you're the least excited about and then progress to those that you're most excited about. This works out especially well if you have a startup that you're very excited about as you can leave it for last, apply, and receive quick turnaround to get into the process.

As for Big Tech, I'm not entirely sure what to think of it. There aren't a lot of job openings, and they will of course will move slower than the hypergrowth startup. Maybe you can apply at the "mid-way" point so they'll kick in when you're doing all the companies at the top of the list?

To be honest, I imagine Big Tech applications are going to be a black hole unless you have a strong referral given the market. If you're pretty set on joining those companies, I hope this in-depth thread about standing out to their hiring managers helps.

To add to Alex's great answer, I'd say that you should also feel free to communicate with the recruiter:

I'm interviewing at a few other companies and would like to align the interviews so I can make the best decision for me and the companies I'm considering. Can you tell me how long your process will take and how much flexibility you have in scheduling?

You don't have to reveal the names of the other companies. Most recruiters are willing to work with you on timeline. Your next job is a big career decision -- you deserve to have sufficient data to feel good about your move.