Begin at the trailhead across from the parking area. The initial trail (Alpine Trail) is a steep step up with railroad ties to prevent erosion, but the trail afterwards is generally well graded. Almost immediately, the trail intersects with another. If you turn right, the trail narrows and will eventually become a single-track trail suitable only for carriers or backpacks. If you do not turn right, the trail remains wide and is suitable for a baby jogger or other large-wheeled stroller, though there will be some rolling hills to push that stroller up!
Continuing straight from the trailhead, you’ll ascend some rolling hills to eventually get to the highest parts of the open space. Old Spanish Trail is largely shaded and winds up and down rolling hills. It will cross Bay Laurel Trail, a nice, but short trail, that would be fun to take with a jogging stroller if Arroyo Trail the subsequent trail needed to stay in the park (versus going to neighborhoods) were not so narrow and steep in places. Stay on Old Spanish Trail until you reach Fire Road Trail, a quite obvious fire road, near where Arroyo Trail and Coal Mine Trail cross. You can follow the fire road or hop onto Coal Mine Trail for a short distance, continuing SSE, passing a water tower.
At the point where Fire Road Trail meets Old Spanish Trail, you can head North down the fire road to reach Sunrise Trail a very short distance later. Walk down the road that Sunrise Trail crosses, to take Los Trancos Trail to your left (a historic trail). This runs parallel to Los Trancos Road and allows you to follow it all the way back to Alpine Road (turn right on Deer Path Trail and left on Sweet Springs Trail, then left on the Alpine Trail parallel to Alpine Road), dropping you at the trailhead where you began. You can also follow Los Trancos Trail and take a couple of other trails left, which will eventually bring you back to trails that drop you back where you began, but with all of the other options, you risk some narrow and steep sections.