working in two states and commuting

From your comment:

My drivers license is from Las Vegas, Nevada, and I'm not registered to vote in either state.

I am working from the assumption that you will be able to prove you are a resident of Nevada and work in California. See California Pub 1031 for Guidelines for determining who is a Resident.

I work full time in California and part time in Las Vegas.

Because of the way California looks at W-2 income your income in California will trigger California state income taxes. Because you are a resident of Nevada, then the Nevada income doesn't trigger California taxes. Of course Nevada doesn't have state income tax.

In California, I have to rent an apartment while working there. I travel to Vegas to work every week and spend some time in the house.

The spending the night in California, and the fact that the presence in the state isn't just for a short duration contract will allow California to tax the California sourced income.

Is my rent deductible?

Generally rent isn't deductible, if you are employee and will be getting a W-2 at the end of the tax year. If you had tour own business and could prove the rent was due to business reasons, you might be able to deduct as a business expense. In some cases where the employer is paying your rent, the amount they pay could be tax free. This would require the reason for staying in the apartment has a direct benefit for the employer.

Is my travel expenses going from California to Vegas deductible?

In general you can't deduct the cost of commuting between your home and work and back. If you had multiple jobs before the 2017 tax changes you could deduct the cost of getting from Job 1 to Job 2 if they were the same day. I don't know how they handle if you don't return to the same place at the end of the day.