Del Valle is a surprisingly fun place to ride, but not very good for beginners or brand new riders. Nearly all the trails are steep, many are loose, and there is a lot of climbing involved.
Brushy Peak is primarily barren fire-road riding. Not quite as steep as Del Valle, but still reasonably steep, and somewhat soul crushing. There is supposedly some fun stuff up near the top, but again it may not be suitable for a beginner.
I would recommend Sycamore Grove park near Del Valle. It's a City of Livermore park, and while it still has some hills, it's much more beginner friendly than Del Valle, and much more scenic than Brushy Peak. You can park at nearby Independence Park to avoid the crowded parking lot and the parking fee, ride into the park, ride up to some of the park's high points to get a view, and if your friend is feeling confident, ride the connector trail between Sycamore and Del Valle. The connector trail is singletrack, but it's mild and relatively flat, so a good introduction to real mountain biking (albeit relatively short). It's easy to piece together a 1 - 1.5 hour ride without getting too bored.