Hi Tom,
You are going to LOVE the scenery! Bring your camera. Lake City is a great place and they allow off-roaders to ride on the streets with no license. Anyway, a nice easy loop out of Lake City is Cinnamon Pass and Engineer Pass. Animas Forks is a cool old guost town and the loop up Picayne Gulch (near Animas Forks) is easy enough too (I think). Even if you only rode up to the top of either pass, you will see incredible scenery. It's big jeep country so intersections are marked well, a GPS is not needed. I'm guessing that the big loop ride might be 40-45 miles. I don't know how far the TTR125 range is, or the range of the rider, but at least go to the top of the passes and back.
My advice for mountain riding is to bring a good enduro/rain jacket and maybe some warmer gloves. When it rains, it gets cold! Also start no later than 8:30-9:00 am to beat the afternoon thunderstorms (unless they are predicting a cloudless day). Plenty of water, food, tools, and waterproof matches or a lighter will do ya. It's a heavily used area with jeeps, ATV's, and off-road bikes, so you won't be alone.
Technically it's not difficult riding in that area, except with the exception of a few climbs on some of the jeep roads, and some of the limited single-track there, but the scenery more than makes up for the less challenging riding.
Have fun!
Kerrie