I had this same problem for so many years - going to the doctor's, having to phone companies, going into banks to sort things out, etc. There was never a clear cut moment where I improved, it was just a gradual shift whereby people weren't around to do these things for me anymore. I think that's what will happen to you too. Eventually you'll be able to do all this without thinking much of it - it just takes time and practice. It's all a case of baby steps though, and you'll never be thrown into the deep end unexpectedly; you'll just slowly start taking initiative as you mature into your 20s.
At the formative level, I think it comes down to parents not letting kids do enough themselves. It's all "oh come here, I'll do it for you" which leaves children with no life skills and no confidence. We develop into terrified adults because of it.
