Yes, I have fished Durango and the surrounding rivers on a couple of occasions. As MassAction states above there is a fly shop right in the middle of town called "Duranglers." Great shop. If you can hire a guide through them to fish the Animas I would recommend it. If he's available I would specifically request Mark Engler, but they are all very good guides. Mark is credited with the creation of the WD-40 fly and he knows the waters up there, and the fly's to use better than anyone in the region (in my humble opinion). My two days fishing with him were possibly my best two days of fishing ever and I have fished with upwards of 30 guides. Heck, Yeti even did a short film about him. He's a Pike guy at heart but guides on the rivers. There is also a very healthly population of Pike in Valecito lake outside of Durango. https://www.youtube.com/watch?v=W71ow-KI2Eg
The Animas is a great fishery but because it's a freestone river and runs right through town so it does get pressure and the fishing can be highly technical.
The San Juan is approximately 1 hour away. I would recommend the drive jst to check it out. One of the most prolific Tailwaters there is.
Duranglers website has techniques and information on all of the surrounding rivers on their website. It's a great resource. There are a lot of small streams in the area like the Los Pinos that are fun as well.
If nothing else I would recommend bringing you fly rod, tying on a small RS2, or WD-40 (like size 22), and when you and the family are out and about, and you see a small stream or river just stop and cast around. www.duranglers.com
Have fun and let us know how it goes...