This is how I would say it:
I prefer going out rather than staying at home.
I prefer going out to staying at home.
I would prefer going out than staying home - this doesn �t sound quite right.
This is how I would say it:
I would prefer to go out rather than staying at home.
If given the choice I would prefer going out to staying at home.
I would prefer going out to staying at home next Saturday.
I would prefer going out to staying at home if the workmen are there making a noise.
I use British English. US English uses �than� differently.