I would like to hear from you folks that have achieved a high-level of proficiency in F# (and also in functional programming in general too) what should be my steps from now on to become a better/professional F# programmer?
I already know much of the F# syntax and have some years of experience with C++. My goal is, as an engineer and mathematician, to design better scientific libraries (linear algebra packages, partial differential solvers, etc.).
you very probably don't want to hear from me as I have not achieved a high level of proficiency with F#, but I have set myself the goal of getting there (much like your self).
I thought I might suggest what has turned out to be my biggest learning tool: Stubbornness.
I set my self a large-ish project (a game, just as gradbot suggests). I then decided to code using as much immutable data as possible regardless of the performance costs.
Then if I couldn't find a way to use immutable data I'd come here and ask for help.
The hoops this stubborn approach forced me to jump through has been a brilliant learning exercise, as (just as Juliet mentions) F# allows you to write really ugly C# if you let your self get lax.
Right now I have tile based 2D world and a little man who can find his way to the treasure and back home again using A* path-finding ... The only thing I mutate is the title of the window it's displayed in.
I only got serious about learning F# towards the end of July (I'd dabbled before then) and this project has taught me a huge amount, along with the help of the guys here at StackOverflow.
The other answers do a good job of suggesting ways to practice your F# writing skills, which is obviously very important. However, something which doesn't seem to have been emphasized much is reading F# code, which is just as important in my opinion. There are a lot of people doing cool stuff with F#, and one of the best ways to learn about the different corners of the language is to read about what they're doing. This will help you pick up idiomatic style and will also expose you to more design patterns, language features, and functional programming techniques than you would be likely to encounter if you were just solving a problem using F# on your own.
The F# blogs syndicated on Planet F# are a good place to start, but there are also several excellent books and presentations on the language.