All of these are possible.
a) I did it!
b) I've done it!
c) I made it!
Tense-wise, the two past tense constructions seem more natural, as they suggest that the difficult journey is now over and done with.
'I did it!' is quite a general statement, and might describe more than the actual journey. For example, it might be the first time that a young person has driven alone on a motorway, so it refers to particular achievement on the part of the speaker.
'I made it' refers to the achievement getting through the journey and arriving home, which is more specific.