I didn't vote for Schwartzenegger but I thought he did a not-bad job considering. He came to office as part of a vicious Republican campaign to destroy the previous governor, Grey Davis. Davis was blamed for a mess caused by his predecessor, the deregulation of electricity that made some electric bills triple. The Republicans spent huge money to convince people this was Davis's fault. Schwartzenegger had the highest name recognition of any Republican leader in California so he got to be governor despite total lack of experience or even agenda.
I don't think Schwartzenegger knew what he was getting into. But most Republican governors of California run as moderates but then once in office they move way to the right. Schwartzenegger did the opposite, running as a Republican and then moving to the center. He ended up losing his support from Republicans while failing to please the Democrats. He faced constant strong opposition from unions and other liberal institutions in the bluest of blue states, California. But very Democrat-like he sought consensus and compromise rather than reflect the unilateralness and arrogance of our president at the time, Mr. I'm-The-Decider.
He ran a special election in 2005 with four initiatives that he endorsed, having to do with teacher tenures, budgetary spending limits, use of union dues for political contributions, etc. They all lost. But he also defeated a popular initiative (in another election) to modify California's 'Three Strikes' law, a change to require the third strike to be a felony. He was also dogged by ethics violations
Most Republican governors of California seem to see the office as a springboard to a presidential run. Arnold didn't because, being foreign born, he couldn't be president anyway. (Richard Nixon, after being defeated by Kennedy in 1960, ran for gov. of CA and made the freudian slip of saying he was running for 'President of California'.)
Anyway it took courage for Arnold to buck the GOP agenda. You never see Republican leaders doing that, not those actually in office anyway.