This mountain has been on my wishlist for hiking for the past few months.  It is just over an hour's drive from our home (Coppet, Switzerland) and because it has a funicular and cable car it is possible to reach incredible heights and views within minutes.  
The other bonus is that there are several walks that can be done downhill - which is always a bonus with children in tow.  Of course, you can walk the trails in the opposite direction, however you would then be going uphill most of the way...and when given the choice, I will save that for when the children are older.  
I was able to sucker (actually, she volunteered very quickly) my friend into joining us as both of our husband's were not available and we have been venturing out finding new things to explore over this Summer school holiday.  She has 2 children as well, so we hiked with a 23 month old, 2.5 year old, 5 year old and 6 year old.  The two youngest children had the best views sitting in their backpacks on our shoulders.
Two moms proud of our accomplishment carrying the 2 year olds on our backs
The children love funiculars and cable cars so our journey up to the top of the moutain via public transportation only added to the fun of our adventure.  The children asked "are we on a roller coaster?" while riding the funicular.
Riding the funicular back down from Plan-Francey to Moleson-Sur-Gruyeres
When we got off the cable car in Moléson we were welcomed by beautiful views and a restaurant.  The restaurant, amply named "Le Sommet", offers indoor and outdoor seating and is open year round.  We took in the views and ordered some hot food before our descent.  There are several seating areas with fenced in areas that are safe for children and there is also an open patio with picnic tables and lawn chairs, however I wouldn't let the children more than a couple feet away from me in this area.
Views from Le Sommet Restaurant

Lucas climbing into a lawn chair to take in another view

Checking out our path back down the mountainside
There are several paths down the mountain from this point.  We originally planned on a longer route that wrapped around the mountain (Tremetta #11), however made a last minute decision to do the shorter (but steeper) route named Bonne Fontaine (#8).  The hikes and map can be found on the Le Moléson website under the Discovery Tab.  Both of the routes depart from the cable car station at the same area and then quickly split in different directions.  My friend and her children are photographed below at the start of the hike before the two trails split.  
My friend, Joanne, with her two children at the start of our hike
Below shows the intersection of paths where the two trails split.  The cows had parked themselves at this intersection which required us to "off-road" a bit to get where we needed to go.
We followed the Bonne Fontaine (#8) trail until it splits and you have an option to head towards a Buvette in Gros-Plané or head towards Plan-Francey.  We opted to go to the Buvette which was a wonderful place to rest our feet, get a drink and let the children play on the playground. This first portion of our hike (from Moléson to the Buvette) on the Bonne Fontaine trail took us 2.5 hours due to the difficulty of descending steep rocks with small legs requiring multiple breaks.  The map says it should take 1 hour, so it took us significantly longer, but we stopped and enjoyed the scenery quite a bit.
Buvette du Gros-Plané
We then continued our hike following signs for Plan-Francey.  The sign said we were about a 30 minute hike from Plan-Francey and it took us just a little longer than 30 minutes with the children.  This portion of the hike was reasonably flat with a short and manageable bout of uphill at the end.

After descending from 2002 meters to 1520 meters very quickly we were happy to being walking on flat land again
 I highly recommend the Le Moléson funicular and cable car.  If hiking isn't feasible, I would even recommend just taking the the cable car up to the summit for the restaurant and views.  They were spectacular and you can check the weather and their webcams before you go to make sure it is a clear day.  You could walk around the top for a bit, order lunch or dinner (or pack a picnic for the picnic area) and then take the cable car/funicular back down. (Or, alternatively take the cable car back to Plan-Francey and walk the last section back down to Moléson-Sur-Gruyere.  Our family has walked up this section before (when the funicular wasn't running) and the path is wide, and was easy enough for Lucas (age 2) and Eliana (age 5) to walk on their own as long as we went slowly.)

While hiking you are often inside the fenced area for the cows.  Therefore, the cows are sometimes on your path.  We loved being so close to the cows, however it was a little bit intimidating for some of the children.  You also have to pass through some of their electric fences.  Most of the pass-through were simple, however there was 1 time we had to crawl under the electric fence to continue on our path.  This was not that simple with backpacks on, so we unloaded, passed the younger children over the fence and crawled under ourselves.
Path goes under the electric fence for the cows.
I would't recommend the Bonne Fontaine path again for families with children.  Instead, I am looking forward to trying the other path (Tremetta #11) as it is longer, but a more gradual descent and very similar views.  There were areas of the Bonne Fontaine path that were so steep the children had to turn around and climb down backwards or slide down on their bums.  I also had to sit down and slide down portions of the trail with the backpack on my back as I didn't trust myself while carrying Lucas.  The children were a little bit nervous and we were too.  Thankfully, we ran into some good samaritans that lent my friend and myself their walking sticks and lent a hand helping the girls (5 and 6 years old) down the steepest portions since we had the younger children on our backs.  Without children I think this path is very reasonable if you are equipped with good hiking shoes and maybe even walking sticks.

Once we reached Plan-Francey, we took the funicular back down to Moléson-Sur-Gruyeres where our car was parked.  We visited the restaurant that offered an indoor and outdoor playground along with several high chairs.  There is also a luge, go-karts and other children activities at this location.


The funicular leaves from Moléson-Sur-Gruyeres and takes you up to Plan-Francey.
The cable car leaves from Plan-Francey and takes you to Le Moléson (the summit).

Hikes start outside the cable car station.

Take the cable car back down to Plan-Francey and walk from Plan-Francey down to Moléson-Sur-Gruyere (parking).

The ticket office for funicular and cable car at easily located on the left side of the parking lot and just below the start of the funicular.  You can buy 1-4 sections depending on how far you plan on hiking.  We purchased a 3 way ticket as we took the funicular both ways and the cable car only one way for this particular trip.

Link to all the photos from this Hike: