I know that I can use either:
You could enter insert mode using I
(capital i).
It will put the cursor at the beginning of the line.
Similarly you can use A
to add something at the end of the line.
Though, it does not really solve the problem of moving while already being in Insert mode.
I have just checked help on Insert mode, there is no key combination in insert mode to move at the beginning of the line.
Other idea : Remap a new command only in insert mode
inoremap