Los Arrayanes National Park is a popular day trip from Bariloche surprise, it's actually located right here in Villa La Angostura. The park is dedicated to the trees that fill it - the red Arrayan tree whose bark looks like curled cinnamon sticks. This forest is also rumored to have inspired Walt Disney's classic Bambi.
You have a few different ways to experience the park - hiking, biking, or on a catamaran excursion. Let's break down your options.
Hike: The trail is just over 11 km each way. Calculate 6 hours to do this nearly 24 kilometer trek. It is very important to check the entrance time restrictions (start before 11 am in winter, 2 pm in summer). There is a visitor center on the isthmus at the trailhead with public bathrooms where you can pay your entrance fee and inform the guides that you'll be on the trail. You can also bike this trail, expect it to take 2 hours each way.
Hike just to the viewpoints: Pro tip, if you don't want to hike far or you don't want to pay the park fee, then you can hike just one hour each way to the Brazo Norte Viewpoint. The park entrance fee is waived for this one particular hike. If you do want to access the other trails and the iconic Arrayan forest, pay your entrance fee.
Catamaran Excursions: Don't want to hike? No problem, go on a 2.5 hour boat tour on Lago Nahuel Huapi and time to explore the park and tea house "Casita de Té del Bosque". It departs from the port in Bahia Mansa, check times here.
Hike + Boat Combination: You can combine the two, hike to the park and take the boat home or take the boat there and hike back. I recommend the latter so that you don't have to stress about missing the boat home. Take the morning catamaran out to the park then enjoy a leisurely walk back, cutting your hike in half.
The above options are how to visit the park from Villa La Angostura. If you want to go on a day trip from Bariloche then this excursion is the best way to go.