If you do a Google search for family-friendly hikes in Auburn, you should find a ton of options. The Auburn area is full of trails.
If you want an easy, relatively flat option, the Miner's Ravine trail in Roseville is a good choice. It's paved and runs through some pretty areas (hard to believe it's in a city). It's also dog-friendly.
Going a little further out, we just hiked to the top of the hill on Angel Island and that's a really pretty hike. You'll get some amazing views of the bay and San Francisco. A lot of butterflies flying around, too. There are two trails leading to the top. One is more moderate (which has the great views) and the other is a lot more family-friendly. Just ask the rangers.