Both of these locations can be seen in a day trip from Ulaanbaatar.
Terelj National Park
When most people think of Mongolia, they often think of the Steppe. Terelj National Park is arguably the best place to see the Steppe. Most people will try and at least spend one night in a ger at the National Park. Given our schedule and the fact that we slept in ger in the Gobi desert, we only did a day trip out to the Park.
Terelj National Park is beautiful, and most people will try to enjoy this beauty on horseback. Unfortunately when we were there, it was raining, which caused us to pass on riding a horse. We did see a number of people riding on horses, but it did seem that they were mainly riding parallel to the road.
There are also some hiking trails, but hiking does not seem to be that popular. This National park has considerable more tourists than the Gobi desert.
Genghis Khan Statue Complex
On the way to Terelj National Park, there is new attraction called the Genghis Khan Statue Complex. While the statute is up, the master plan is build a huge complex with hotels and other attractions.
It is a great photo opp and worth spending an hour there.