Here are my opinions and please note they can be expanded upon and generalities so there are exceptions of course. I grew up in Hamilton (in the NE but lived everywhere), moved to Toronto then moved back to raise a family. I love living in Hamilton and think that it is a great value.
James Street divides the city in East and West and King Street in North and South (not counting the Mountain which is its own thing). Basically, the factories are in the NE and the closer you get to them the less desirable the area. If you have North on your address you will likely pay $20K less for a home on the same street. There are some pockets of really great housing stock in the East End (like St Clair Neighbourhood or the Stinson neighbourhood) but a lot of the bigger homes were duplexed so if you have a single family home that you bought for $200K you may live next to a house that has 4 apartments. Also, with Stinson, well I wouldn't feel comfortable living there. The schools are not great l as there are a lot of behavioral issues. Also you will have to drive everywhere as the areas are not very walkable with the exception of the Ottawa Street strip and the arts section on James Street North (which is essentially downtown).
The south west side of downtown is great (area boxed between James South and Dundurn and Main and the Mountain) and the closer to Locke Street the better as it is the commercial district here and the housing stock is really good but $100-$150K higher than the same house in the East. Even the other side of Queen can bring the same house in lower. Locke is the trendy area right now. I chose this area because it was walkable, the schools are good and it is close to the highway and commuter trains to get to Toronto. Westdale is near the University and was one of the traditional money parts of town but is now getting overrun with student housing.
The "Mountain" (living on top of the escarpment overlooking the city) is like one big suburb. If you like driving then this may be your area. Personally, I never considered it at all when I moved back to Hamilton.
Dundas and Ancaster (also in the West) are now amalgamated with Hamilton but used to be small towns. I like Dundas better and it is a super cute area but pricier. Ancaster is where you live if you make big money.
To look at schools I went on the Board Of Education site and clicked the school websites then looked at their EQAO ratings. I also looked to see what activities they engage the kids in to make up what I considered a good school. I have met a few teachers here and more or less their working experience has borne out my instincts.
All in all, Hamilton has a lot of beauty in it. The factories are cool looking but most people think they are ugly and it is the skyline you see when you travel from Toronto to Niagara/Buffalo. If you go into the West End there are beautiful lakefront parks where you can canoe or hike and that is the hidden Hamilton that nobody knows about. I have a few other posts on Hamilton in here if you are interested in knowing more or have specific questions let me know.
Some pictures.
Hamilton in Pictures | Reader's Digest