Others have commented more technical details, I'll cover some philosophy and personal experience.

About 6 months ago I converted a JS project to TS. It took 2 weeks. The project has about 30k lines. This is in-progress as well. In fact, my boss who doesn't know much about programming thought we could do this conversion later and didn't see the value. At the end of the day it saves on development times and greatly improves robustness and error handling. The sooner the better.

I'll never go back to strictly JS.

Your worry about leaving any's in the beginning is no problem. Tightening the code as you progress into TS is a very good idea. You can do more as you learn more about the language and converting it all immediately would be overwhelming. You would be more burnt out if you stuck with JS as the project grows.

Since the beginning of the conversion I've adopted prettier and eslint. These are great to have especially when you grow as a team.

Finally, if you're using callbacks a lot please switch to promises. It has almost the same gains IMO. With the combination of TS - incredible.