I had a place in Los Feliz for $850 about 8 Years(!
) ago. That had good riding with easy access to Griffith Park and the LA Bike path. Plus the 134 /101 gives fast access East for Pasadena / San Gabriel Rides and the 101 for San Fernando/Santa Monica Mountain rides. No idea what the rent would be like these days.
Burbank would also give you good access to these areas.
If you're in Pasadena, you can take the gold line train into downtown and avoid the hellish traffic. I think there's a train from the Los Feliz area to downtown also, maybe that's hollywood though?
Either way, you'll remember mountains quickly. You just can't avoid them out here, nor would you want to. They're the best riding LA has to offer since they're the only way to avoid traffic.