Not sure about the rentals, so follow peckma's advice. And yes, BH is a cycling black hole. Be sure to stop at every stop sign because the cops there seem to target cyclists.
BH, however, is small, so you can escape easily. Going west is the beach and PCH. Try to hit the beach south of Venice or north of Santa Monica. The Ballona creek bike trail is your best bet for the south side, Venice Blvd is also somewhat bike-friendly. I would suggest something like the donut ride
which takes you up and around Palos Verdes.
If you want, you could also head north and climb Beverly Glen up to Mulholland and over. Definitely more traffic on these stretches. You could also take Wilshire to the beach and take PCH north of Santa Monica. It is traffic-dense for about 5-10 miles, then clears up past central Malibu. Lots of cyclists on that route, so you could probably hop on to a group for extra visibility. Past Malibu, you could keep going or take a right and hit any number of canyon climbs. Kanan and Las Virgenes are two, there are more.
If you want serious climbing, come east. Take the Metro, hop on the Expo line in Culver city and navigate your way to the Gold Line. Get off at the end in Sierra Madre and ride out to the San Gabriels for Glendora Mtn Rd, Hwy 39, and more. These are pretty long and epic compared to the west side climbs, but also much more traffic-free.