You're going to have a hard time getting feedback on entire neighborhoods. Durham is so small/close-knit that anywhere you are you won't be too far from the 'bad areas.' IE - you may live in SW Durham which is very nice, almost rural/suburban and you're still not far from the West End or Lakewood areas which aren't as nice... (they're not horrible though.)
What I can tell you:
Old West Durham - nice neighborhood, very safe but also very expensive.
The same can be said for the Hillandale area, it's a little more suburban too.
Trinity Park is one of the nicest looking areas in Durham, just IMO. Beautiful homes, it's very close to everything and it has much more of an urban feel generally. But, there are also some bad areas to the Trinity Park neighborhood (which is sort of vaguely defined). It's one area where the "have's" and "have-not's" are obviously visible. You can drive down Markham and coming from the East you'll pass through 'rougher' looking neighborhoods until you get close to the East Campus and you'll be like, Jesus - these are the nicest houses I've ever seen.
Regardless, you should be safe. I would avoid some areas around Washington St, Buchanan has rough spots, Macon St etc. Driving around you should quickly get a feel. I grew up off Monmouth Ave when I was younger and never had any real problems.
Old North Durham is a similar story to Trinity Park, except that there are more ghetto parts to that area. I'm not slamming the area because I would rather live in the worse off parts than somewhere like Hope Valley or some other suburban community, but if you have real money to spend then you can really get a nice spot. Roxboro St and Mangum are the two 'main roads' that go through that neighborhood and you can get a feel for that there... I would just recommend staying away from Geer St, Dowd St & Canal St. I'm not trying to scare you away because the 'bad' of Durham has always been overblown, especially considering how nice it has become recently, but you just need to be conscious of the good and the bad. Old North Durham has a great vibe to it and I would really recommend checking it out.
Now... for Duke Park.
Duke Park is an interesting neighborhood and I really don't know how to explain it. I live in what is considered the 'Duke Park' area and I get some news letter that I don't really care for, but I have never heard anyone from where I'm 'from' at the moment, claim to live there.
It's the same old story in the Durham in that there are some very nice, almost yuppieish areas to Duke Park. You'll find that off Markham. There are some very, very nice houses and it is a predominantly white area. This is where you'll find people with Duke Park Neighborhood stickers on their car (and silly signs about shooting speeders with paintball guns

)
Now, when you head a little further east, you will hit Avondale, which is what I live off of, and that area tends to have a worse reputation. It's predominantly black and poor and there's more crime, drug problems, people wandering around, police presence etc and all that. I'm just being honest here, not trying to scare you away, because I actually really enjoy living here and like anywhere in the world, if you're smart and mind your business you generally won't find trouble. It's just something to consider because of your family. It is night and day from the area off Markham though.
Hope that helps!