Well, in Denver itself there is the Platte Valley Trolley (although I recommend checking their website as it doesn't operate all the time, but I believe that it is right now) and the Forney Museum of Transportation. They have a range of various transportation vehicles with some good train pieces (one of their main railroad pieces is UP Big Boy 4005).
About a half hour from Denver is the Colorado Railroad Museum in Golden. If you can only go to one place, I would recommend that it would be here. It is the largest railroad museum in the state. They have around 100 pieces of rolling stock ranging from small narrow gauge cars to big standard gauge steam locos. Definitely a good place to spend a few hours. It has a collection that is good representation of Colorado's railroads. If you happen to go on a Saturday, that is the day they run trains, and it is typically one of the Rio Grande Southern Galloping Geese, but they do occasionally fire up one of their steam locomotives.
Model/Toy Train wise, you could stop in Caboose Hobbies (in Denver of course), the self-proclamed "largest train store in the US". They do have a very large selection of model railway products, I can say that.
About 45 minutes or so is Colorado Springs, and something to do their is visit the Pikes Peak Historical Street Railway Foundation's museum (I myself recently discovered this museum, and I have lived in Colorado for 15 years.) They have a small collection, but it has quite a few trolleys as well as some regular railroad rolling stock. Another is the Manitou & Pikes Peak cog railway in near-by Manitou Springs. It is a classic cog railway that winds its way up to the summit of Pikes Peak.
About an hour or so from Denver (depending on where you are) is Fort Collins and there they have the Fort Collins Municpal Railway, which is a short trolley ride through town. Another attraction is BNSF's mainline which runs down the middle of Mason Street.
These are the only things that I can think of that are within an hour's drive from Denver, most other railway attractions (such as the Rio Grande Scenic, Cumbres & Toltec, Durango & Silverton, and the Leadville, Colorado & Southern) are at least a two hours drive.
Hopefully this helps!:wave: (I would recommend all of these, but I'm sure that there isn't time to visit all, so if I had to pick one, it would be the Colorado Railroad Museum.)