Cheyenne Mountain State Park right across the road from the Ft Carson main gate and is the best place for a new intermediate to break himself in to CSprings Riding. It's $7/visit or $70 for an annual pass, but will be well worth it if you're living on/near post. The climbs are moderate and the trails range from wide and easy to narrow an techy, but nothing that would cause mortal harm.
After that, it's worth a trip to Ute Valley Park and the Red Rock Canyon open space. Then, once you feel like you're getting your altitude lungs, head up to the many trails in the Cheyenne Canyon area (Buckhorn, Captain Jack's, Columbine, Section 16(the latter is more of an expert descent)). Cheyenne Canyon is alwo pretty close to post.
Palmer Park is a great place once you feel comfortable with the techy stuff. There are easy trails there too, but it's got so much gnar that it's best when you're comfortable with lots of rocks, both up and down.
The Falcon Trail around the USAFA, is a nice intermediate ride. There's also a nice intermediate network on the west side of Monument, and Spruce Mountain just off the interstate on the way to Denver is a good beginner/intermediate ride with nice views of the Monument Valley and Pikes Peak.
The nearest "must do" intermediate ride in the mountains is Rampart Reservior by Woodland Park. There are a couple rough spots requiring a dismount, but the 14 mile loop is mostly intermediate, has great views and no major climbs. I'm not sure how the fire affected this area however. Maybe somebody else here has been up there since the burn.
Once the snow flies, you're less than an hour from Ft Carson to Lake Pueblo State Park (the same pass for Cheyenne Mountain is good here). This area is further south, lower in elevation, and stays much drier. Once the heat passes, head here for both long, cross country and short downhill style rides--you can do either/or both in a day.
After that, you've got literally hundreds of possible routes within a day trip.
Congratulations of a great assignment!