There are quite a few trails that would probably work, however, without more info it would be difficult to recommend any particular one. I would suggest you do a google search for "small child friendly hikes santa monica mountains". Then pick one or more of the hikes that sound good to you. I would suggest you drive up highway one, the Pacific Coast Highway along the beach. There are several hikes you can reach off of the Highway.

Greetings. The Sandstone peak hike (the Mishe Mokwa trail) is a superb choice. I can’t include links but Google search “Mishe Mokwa” and look at the “modernhiker” page. Copy this into Google maps search box: 34.1131337 N 118.9260686 W and you can check out the trail. You’ll notice that the trail splits off to the right a short way up from the parking lot (this is the actual trail). My advice is to take this split and go up this way; it’s a better “up” hike and coming down the Backbone Trail is a better “down” hike. Take water and a camera. As to where to stay, I recommend Santa Monica and best bet is on/near Main Street in the Ocean Park Section; you can easily walk to the beach and a variety nearby of restaurants; the Sea Shore Motel on Main that’s worth a look. While Malibu is beautiful, there’s not a lot to do nearby and it’s pricy; avoid “LA” unless you’ve got some bucks then try the Bel Air Hotel or the Beverly Hills Hotel or the Beverly Wilshire. Santa Monica has a good bus line for local getting-about. Hope this helps.

Tricky...The Santa Monica Big Blue bus route 9 from downtown Santa Monica can take you to Temescal Canyon Gateway Park which has a lovely, though well-traveled set of trails. If you Google map "Temescal Canyon Pacific Palisades", hit "Directions" and click on the bus icon, you can see the bus stop at the entrance to the park. It's less than a 1/2 mile walk (a nice one) up the canyon to the trail head. Google "Temescal Canyon Gateway Park" which takes you to lamountains com/parks and you can better see the layout of the park. Off hand, I think that's the only public transit access point to trail heads I can think of. Hope this helps.

You can take Topanga Canyon Road to the quaint town of Topanga. Several roadside shops there. The Inn of the Seventh Ray is a very romantic restaurant, though rather pricey. Not a lot of roadside attractions in the Santa Monica's. From Topanga you could go up Fernwood Pacific Drive. At its crest go right on Saddle Peak Road. Just past its summit there is a view point for viewing into the mountains and back toward the ocean and the LA Basin. From there you could head back on Saddle Peak, and turn right at Fernwood, which soon becomes Tuna Canyon Rd. This is one of the wildest, steepest canyons in the Santa Monica's. You'll need good brakes. For the best view of the Ocean you'd need to travel 20 miles north on Pacific Coast Hwy to Deer Creek Road. Up along this road are great ocean views. From there you could head further into the mountains, then left on Yerba Buena. This is the back woods of the Santa Monica's. After a few miles you will come to a ranger station called The Circle X Ranch, where there are a couple of picnic tables, and views of Sandstone Peak, the highest point in the Santa Monica's. Ask rangers about hikes around the ranch. From the ranch you could continue along Yerba Buena, which eventually becomes Little Sycamore Rd, which ends on Mulholland Hwy. Turn right to go back to land on PCH near Neptune's Nest, a popular coast restaurant. Or turn left on Mulholland for more miles of mountain driving. Eventually you will come to Old Topanga Canyon Rd., which you could take back through Topanga to PCH. There are many canyon roads going up from PCH. Maybe someone else can come up with some more good suggestions.