Hey Justin . . . can't say I know the Park Slope of NY, but I do know the areas of Portland.
And getting down to the brass tacks . . . it's all about what your price range is. Unfortunately, or practically speaking, it does boil down to financing and what your family's monthly budget will be comfortable with in a mortgage payment. Even if you're paying cash, you still have your price point and that decides what you can purchase comfortably . . . and where.
From reading some of the answers here, I get a good idea of what Park Slope is all about. The thing with that is, the areas that would be more "Park Slope-like" are in the Downtown area of Portland. Downtown neighborhoods, such as the Pearl, Alphabet Blocks are going to be spendy . . . higher price points.
There are some great city-feel neighborhoods (as opposed to suburbs). Check out BuyersAgentsPortland.com .the "For Sale By Neighborhood" tab and mouse over the the various neighborhoods of Portland -- check out the neighborhoods of NE Portland!! You can look at current listing and get a feel based on price range.
The other questions would be, are you relocating here for employment? Where will you be working and what's the commute like? The twins will be elementary school ready before you know it . . . find a good school is very important too!! Lot's of things to consider in a move.
I help a lot of relocation buyers acclimate to the Metro Portland region. I have helped buyers from NY if that gets me any bonus points!?! ;-)
Carla Muss-Jacobs, Principal Broker/Owner
EBA Portland, LLC
Exclusive Buyers Agent since 1999
The Web Reference has a nice little home in the Alberta Arts neighborhood: