Downtown? Sorry, but all the best options will require you to ferry you and your bike to the start regions away from downtown. Your choices are either head to the coast where you'll have a 25 mile long bike path (best riding is during the weekday), or the Griffith Park region (weekday, and early mornings). Then there is Mulholland Dr which runs through the Hollywood hills (Santa Monica mountains) beginning where the 101 crosses through the mountains. Starting in Santa Monica (or any coast city) you can head up to Malibu (partly along the beach path, the remainder along PCH) and enjoy the Santa Monica mountains. You can also head north to the San Gabriel mountains (Pasadena) for some great climb rides.

Check out a map of the region and familiarize yourself with the geography, then ask specific questions.