Berwyn is a nice area too and is really working on improving its image but as others have said the schools are not the greatest and when we looked there we found a lot of houses with a ton of "bedrooms" that weren't legally bedrooms. But there are some beautiful bungalos there but we ultimately found that the layout of the standard bungalo didn't work for us (it's generally two bedrooms on the main floor with a bath in between and then an attic conversion (complete with short ceilings) upstairs. We ultimately abandoned Berwyn as a possible location for us because the future of the schools is too uncertain and we didn't want to risk our resale value.
La Grange has reasonable houses just avoid the super pretty downtown area. It's gorgeous but you're looking at housing upwards of 600k and into the millions. La Grange Park has all the amenities with some more reasonable prices. We saw a gorgeous rehab in La Grange Park, 4 bds/3bths for $380k which was honestly a bit overpriced. We would have bought it but we were waiting for our condo to sell.
Depending on how long you plan on being there look into the high schools. Both Riverside/Brookfield High School (Brookfield, Riverside, North Riverside, and parts of La Grange Park filter into it) and Lyons Township (La Grange and a few other suburbs filter into this) have excellent scores but will offer very different experiences due to their sizes. Lyons is HUGE with two campuses and they split the freshmen and sophomores from the juniors and seniors.
Honestly, house hunting in the area was very trying. We loved Oak Park but the houses in that price range often needed a lot of work and/or were just really lacking in space. Unfinished basements with low ceilings are very common from what we saw. Houses in Oak Park like what we wanted were still 500k+ with very high taxes (they take their high school seriously but with somewhat mixed results - OPRF scores lower than both Riverside/Brookfield and Lyons but is still excellent). Elmhurst had more updated houses with more space but the ones in our price range (same as yours) were often in a less desirable location (right near the highway/train, etc) and were right at the top of the budget). Riverside had some nice stuff too but again, higher priced because it has that notoriety (Riverside itself was designed by the guy who did central park) and they seem to pay the lion's share of taxes towards the high school over Brookfield.
We saw a lot of homes that were overpriced considering the condition they are in. A lot of outdated kitchens and baths yet sellers seemed to think that it was worth what it would be if the kitchen and baths were newer. Lots of 70's white cabinets with the wood strip at the bottom. And sellers just don't seem to get it. It's understandable, I was a seller myself at the time, there's no motivation to update/upgrade because that's money they won't see back, but they still think their homes are worth more than they are. But there are some gems out there and I like to think we got one.
The forums on city-data.com are also very helpful and helped us narrow down our search.