What neighborhoods in Colorado offer the best public schools?

4G Denver native here, I still live where I grew up. So if you're looking at Denver you'll probably want to stay in southeast Denver for your needs. Easy access to I-25, lots of families, safe, good shopping, and the best schools.

University Park is indeed an excellent school. I lived across the street from it for 4 years. Specific neighborhoods you'll want to check out in Denver are: University Hills, Washington Park, University Park, Glendale, Virginia Village, Cory Merrill, Belcaro, Washington Virginia Vale, Cherry Creek, and that area. Congress Park is nice but very urban compared to the neighborhoods I just listed, which are more semi-urban and will offer the good parks and schools you're looking for.

Washington Park is basically to Denver what Central Park is to New York City. A quieter neighborhood that is home to University Park Elementary is University Park and they have a large green park called Observatory Park. Lived there for years, it's LOVELY. Down the street from DU and easy access to most everything.

Here is a link to DPS' School Performance Framework report if you're looking just at Denver: http://spf.dpsk12.org/documents/current/05SPF%20-%20Rating%20and%20Indicator%20Summary%20(Traditional).pdf
Here is a link to Colorado School Grades Elementary Schools if you want to expand your scope: http://www.coloradoschoolgrades.com/SchoolSearch.aspx?topRanked=true&st=1

I personally went to University Park Elementary in Denver which is an exceptional school in DPS SPF ratings and a B+ school by Colorado School Grades.
The neighborhood is currently being gentrified. Every house that is sold gets scraped and then a two story house is built, however there is affordable housing and apartments in the neighborhood. The Harvard Gulch Trail cuts through the neighborhood and is a great place to walk. There also are two main parks: McWilliams Park (Dino Park) and Observatory Park (O-Park) We're not far from a library and lots of shopping places nearby. There also is a YMCA that is currently being remodeled. Overall, its a charming neighborhood with a nice school and a nice environment.