Hi Catherine,
I like - central Bendigo (walk to fountain/shops), Kennington, Quarry Hill, parts of Flora Hill, Golden Square walking distance to town (but not all of Golden Square), and areas within a few blocks (walking) to the hospital (parts of Bendigo North near the lake).
I'm not keen, generally, on Long Gully, Cal Gully or Kangaroo Flat, though these are where many of the cheaper properties are.
Eaglehawk also has a bad name, but I think this is changing. I know quite a few professional people buying properties around the lake there (that has recently had an overhaul) and doing them up. So around the lake and a block or two from the main street of Eaglehawk would be worth a look, imo.
I'm not familiar enough with the family estates located away from the city centre (inc Spring Gully etc) to comment on them.
Whatever you do, make sure you have a building inspection "to the buyers satisfaction" as part of your contract when purchasing. Many of the beautiful period homes that make Bendigo so attractive, also have a lot of hidden problems.
Best of luck with your search. Please let us know how you go.
Caroline