This route is either an easier summer hike or a tougher winter hike. In summer, many people prefer the Höher Gang instead of the longer, slightly tedious Fahrweg described here - however, in winter the Höher Gang is not accessible! Check out other routes in this area if you want a longer summer hike.
The start of the walk is slightly tedious - a long section through town and then up the side of a ski slope. But once you get away from the skiers you'll have a great time. The route loops up onto a high cliff, and the path passes through woods under steep cliffs. It's incredible to look up at the mountains towering above you, including the impressive Ehrwalder Sonnenspitze, the "Matterhorn of Ehrwald".
As you round the cliffs you can look up into the valley which contains Seebensee and Drachensee. After a short, steep climb you reach the Seebensee, which in winter is white and flat. Perhaps pause for some lunch or a mushy banana. The next section takes you down into a small valley, under some impressive cliffs, and from here you can see the Coburger Hutte, your goal for the day.
The next section is very steep. In snowshoes and good snow, the best bet may simply be straight up, keeping slightly right before cutting back on the level to the hut. The hut is closed, but pause to marvel at the mountains around you, the expanse of Drachensee and the Zugspitze rising over the far side of the valley below. Here this route turns around - the ambitious might try Drachenkopf, but it requires proper crampons (not spikes!) and some exposed scrambling along the ridge. If in doubt, just savour the view a while longer and then return - Drachenkopf will be there another day for you!
The way back down is a bit of a slog as there is a long, monotonous uphill section before you can drop to the ski slope. Maybe reward yourself at the bar at the bottom of the ski slope - you can get a ball of steamed dough, covered in sesame seeds and swimming in butter. Delicious!