I really like Ivanhoe and the surrounding areas (Eaglemont, Heidelberg, Fairfield, Alphington). It's a beautiful area, LOTS of green space, and excellent public transport (REAL mass transit, not stupid ugly trams that hold you up when in the car, and make you slip on the tracks and fall off your bike in the wet!! I hate trams, does it show? :o )
And best of all it's the closest thing you can get to a central cycling hub for the city, no matter what your discipline 😄
You want road riding? Kew Boulevarde is just down the road (or Ivanhoe Boulevarde for a nice loop without any real hilly stuff) and the hills of Eltham, Research, Warrandyte etc. aren't all that far away. For long flat rides, it's only 10km into the city and 15km to Station Pier to start a bayside ride to Mordialloc, Frankston and beyond.
Track? Guess where the CW games track cycling velodrome is... 😉 (any ACF licenced rider can train there)
MTB? Yarra Trail singletrack is pretty much at your front door, Mt Disappointment and Kinglake is a relatively short drive to your north, or for Lysterfield just go down the South-Eastern
For recreational riding, you need only ride a short distance to link up to pretty much every bike path in the Melbourne network. Eastern Freeway, Main Yarra Trail, Darebin Creek, Capital City Trail, Merri Creek, Gardiners Creek/Scotchman's Creek, Anniversary Trail, they're all very easy to get to.
The downside is that buying a house is an expensive proposition (in some areas really excessively so), but as with any region there are pockets of better value.