- This sport is suitable/fit for me.
- This game is too violent. It is not suitable/fit for me.
- I moved to south. The climate there was not suitable/fit for me.
- This woman is suitable/fit for me.
- Teaching is suitable/fit for me.
As Neil notes, it is more common to say "this doesn't suit me" than "this isn't suitable for me". "Suitable" is more formal.
"Fit" implies a level of quality or ability. If you say, "Sally is not fit to be my girlfriend" you're saying that she's not good enough. That would be an insult. But if you say, "Sally does not suit me as a girlfriend", it's more ambiguous. The issue may be that she's not good enough, or it may just be that the two of you are incompatible. Still, I'd avoid using either term to describe a person unless I didn't care about insulting them. Instead I'd use a different choice of words that are more clear, like "Sally and I were not a good match."
Also note that "fit" can also refer to something being of the appropriate size or shape. Like if you say, "This shirt doesn't fit me", that would normally be understood to mean that it was too big or too small, rather than being a style that you don't like. "This shirt doesn't suit me" would be understood to mean you don't like the style.
Using suitable as you have is correct, though it tends to sound better to say "It doesn't suit me." rather than "It is not suitable for me."
As for fit (guys across the pond correct me if it's not this way in Great Britain), it tends to be more used as a gerund, as in "This cocktail dress is not fitting for a formal event." Used as an adjective, the meaning tends to lean more towards "in shape" than "suitable," as in, "You're looking fit. Did you lose a few pounds?"
So, as in your sentences you add "for me" (particular person), I would say(1):
(1) However, it would need more context.