I wrote a post on the different neighborhoods of Duluth awhile ago. You can find it here:
http://www.city-data.com/forum/minnesota/127352-why-duluth-great-place-live-post2014103.html#post2014103
I admittedly focused more on East Duluth neighborhoods than West Duluth neighborhoods in the above mentioned post, but that's not to say there aren't any nice neighborhoods out that way. In your price range in East Duluth you will be a bit limited, but still could get lucky in Lakeside, Kenwood or Duluth Heights. You could also get a steal on a older home (probably a little bit of a fixer upper) in lower Woodland and Congdon if you look closer to 21st Ave East.
You'll have many more options in West Duluth. If you really want to get out of the "city" but still be in the city limits, you could check out Gary New Duluth, Morgan Park, Riverside, Spirit Valley, etc. Gary New Duluth is the farthest community West and is fairly safe. I'm not as familiar with West Duluth as I'd like so I can't really offer much advice on neighborhoods there, other than I would recommend avoiding lower Lincoln Park. Another area to avoid in general in Central Hillside (just above downtown).
If you're looking to rent a house (or duplex), rent usually ranges from $600 - $1000 for a 2 - 3 bedroom in Lakeside, Kenwood and Woodland. If you want to avoid college rentals, checkout Lakeside, upper Woodland (around Piggly Wiggly) and Duluth Heights. Areas around UMD (in Woodland) are obviously heavy college rental areas and you will find plenty of large, run-down and abused college party homes.
If you don't mind renting an apartment or townhome, you have many options. There are apartment complexes in most neighborhoods. There are also plenty of townhome developments around the area.
Check out Bowman Properties (
Bowman Properties, Inc. - Home), Melhus (
Property Management Duluth Minnesota - Apartment Rentals Duluth Minnesota - Melhus Management Company) and ShipRock Management (
Welcome to ShipRock Property Management). Bowman especially has many apartment complexes and townhome developments. There are other developers/management companies here, but those are a few of the larger ones.
Good luck on your move here! What brings you to our city?