The Denver International Airport is the closest airport to the area. The Denver Airport Train will get you into downtown Denver really easily. Alternatively there are many option to rent a car or Uber from the airport.
You can stay anywhere in the Denver area, transportation is easy from anywhere in the city. The Uptown, Capital Hills, Downtown, and Five Points neighborhoods are close to the venue and have a lot of nightlife and restaurants to explore.
The weather in June can be anywhere from the 70-90 degrees. It will be sunny and the ceremony will be outside so plan for warm weather.