My family of 4, including 2 teenagers, just spent 4 very peaceful and enjoyable days in Red River, NM. Here's my summary:
July Weather - Highs in the low to mid 70s and lows around 40. Some overcast periods with afternoon mountain showers but overall absolutely beautiful. Wear sunscreen - the sun is INTENSE.
Altitude Sickness - The town sits at 8750 feet, the pass at 9850 ft and the top of the ski are is 10,350 feet. Drink LOTS of water and eat carbs (fruit is great) to avoid the nausea, dizziness and headaches common with altitude sickness. It will pass in a day or two as you acclimate.
Town - You can walk from one end of town to the other and it's a very safe town for the kids. Lots of family activities. The town community center is very active and offers lots of free family and kid-oriented activities. There is a limited choice of restaurants and most close by 9 pm. We tried Capo's and Sundance and would recommend both. Calzones are great at Capo's and Sundance has great green chili. There is one grocery store (pricey) and 2-3 bars with live music on the weekends. There are many shops selling the typical tourist stuff.
Lodging - Lots of ski condos and accommodations of all types. My recommendation is to get something with a kitchen and bring as much of your own food as possible to save money. Many places are pet friendly and lots of people bring their dogs.
Activities - Very outdoorsy area. I highly recommend riding the ski lift up the mountain and hiking around up there. The views are great, there's a small restaurant, and even a frisbee golf course on top of the mountain. There are a handful of businesses that arrange trips and rent jeeps, mountain bikes, digglers (2 person motorized vehicle), surreys (multi-person bicycle with awning), mountain boards (like snowboard but with wheels for grass) and ATVs. We rented a jeep through Sitzmark Sports, but they all seem to broker for the same place and have consistent prices. We couldn't find anywhere in town that rented ATVs for your own use, they were all guided tours. White water rafting is about an hour and a half away so we decided to pass and stay in town but you can book rafting trips through the local businesses. There are tons of hiking trails in the area. Lots of fishing opportunities and the local river is heavily stocked with trout. Red River Stables is a great place to ride horses.
4x4 - The most popular activity seemed to be 4-wheelin'. There are lots of jeep and ATV trails. You can rent a jeep locally but you'll need to bring your own ATV if you don't like the guided tour thing. These are serious jeep trails and I would not recommend taking your own 4x4 up some of them. The trails up through Mallette Canyon up to Midnight Meadows and Greenie Peak are fun. Old Red River Pass has great views from 9850 feet. The trail to Cabresto Lake is well worth it, but make sure you have good ground clearance. The trail to Goose Lake is fantastic! It was a major white-knuckler - steep, narrow with drop offs. You start the trail by pulling off the highway and fording through the river - very cool! In June the river will hit the hood of the jeep. In July it was above the tires. The trail is very rocky and feels much longer than 8 miles. The trail is narrow and in many places oncoming traffic becomes a real problem - just watch out. There are old mining cabins and caves/mine shafts on the trail. The top lets out at Goose Lake, which is gorgeous. Our favorite trail by far.
Wildlife - There are lots of elk in the area and tons of chipmunks.
There was a black bear waiting to greet people on the Goose Lake jeep trail. We saw a snowshoe rabbit, lots of birds (grey jays, mountain blue jays, bluebirds, ravens, hummingbirds, ducks), and several deer. The deer and chipmunks are used to people and are often wandering around through town an in your front yard.