if you want to be near the city but not in it, you could also consider coastal areas. The advantage there is being near the ocean for summer months and having the Santa Monica mountains for riding. The valley side (Burbank etc) gets exceptionally hot during the summer - a temperature difference of 30 degrees between the coast and inland isn't uncommon during the summer. THere's also an issue of air quality.
You mentioned you'd be renting. Malibu has some rentals that are around the same ballpark you'd find in other places in LA. On the other hand, if you want to be more around an urban experience, close to nightlife etc., and inland you should consider Los Feliz, Silverlake. Nearer the coast would be Santa Monica area.