One good approach:

1. Begin by typing all your models - interfaces/types/classes - as this should be the source of truth for types. While doing this, avoid (where possible) to use the any type.
2. Type your functions - Although most times, the language service/compiler may infer what types come in and out of the function, adding types will ensure your compiler tells you when you're not using your functions correctly
3. Use Generics to your advantage - Angular has many functions that accept generic types that you can leverage to ensure you're getting the expected type. One good example is the HttpClient's functions: get, put, post, delete. Say, you're expecting a number type from an api call, then you'd add they type to the function invocation: this.http.get<number>(someAPIUrl). Most of the libraries have good documentation on their typings.
4. You can start turning on the tsconfig flags such as noImplicitAny - this will help narrow down the places where you need to add more typing that you may have overlooked. At this point, you can even turn on strict mode as well.
5. Turn on template type checking - The Angular docs are super helpful for this - I can't tell you enough how easy it is to click on an object's property in the template and it takes me straight to the class/interface/type definition.

Of course, it may take some time before you get to where you want and expect some things to break before you fix them. So working through it bit by bit my help.

Just to note, there's more work that the Angular Team is doing to improve typing such as Typed Forms.