There are tons of free-outdoor things to do in and around Denver.  You can head up to Red Rocks Amphitheatre-the area is beautiful, and you can get a good workout in running up the stairs.  Take in the adjacent town of Morrison for lunch or dinner. 

Head up to Boulder and walk around Pearl Street Mall before hiking Royal Arch.  Plenty of brew pubs around here too.

In Denver itself, there are tons of local breweries-downtown and in the Highland area.  The Denver Zoo has a brand new, excellent exhibit for elephants that is very cool.  The Denver Museum of Nature and Science is wonderful with travelling exhibits that change throughout the year and wonderful i-max shows.  I also am a big fan of the Denver Botanic Gardens. 

Lastly, grab a bike or you walking shoes and head down to Washington Park, take a local brew and soak up the sun!

Take a walk around the pond in City Park and walk up the hill toward the Denver Museum of Nature and Science, on a clear day you'll get a good view with the skyline, the mountains, the pond and the late-1920s boathouse/community building all in one. 

Confluence Park is a good place to sit and people watch on a sunny day. It's the point where the South Platte and Cherry creeks meet. 

Check to see if there are free days at the Denver Art Museum, they will periodically open their doors for a day once or twice a month, especially during the summer. 

Scrape together a few bucks and get ice cream from Sweet Action Ice Cream, Ice Cream Riot or Liks Ice Cream. 

Try looking at Westword www.westword.com and 5280.com, http://www.5280.com/cultureandevents/digital/2015/06/10-mostly-free-things-do-june-2015-edition (I think they have past months too).

Red Rocks Amphitheatre, 16th Street Mall, Denver Art MuseumCity Park (Jazz in the Park on Sundays), Denver Museum of Nature and Science and American Museum of Western Art--The Anschutz Collection and Museo De Las Americas.  I believe all those museums have free days go to their web sites.  

Also Art District on Santa Fe has first Fridays (of the month) open galleries.  I think River North Art District has started something similar to it on another night. 

Break the bank Coors Field and get rock pile seats the day of a game for 4 bucks (at the box office and King Soopers) go early scalpers will sell them for about $8-10.

Try farmer's market's.   

The Denver Mint is always interesting. Tours fill up quickly so call ahead. Also check out www.milehighonthecheap.com/