The route starts at the Les Planards ski area. Walk up the piste taking the blue run that leads off to the left, after about 100 metres or so the run will open up again to the right. Here you can choose either to carry on to the left on a shallower route to the Mer de Glace that passes via the Rochers des Mottets refuge or take the steeper trail which starts half way up the section of piste to the right, entering the forest on the left hand side. The steeper trail leads through the forest towards the arches of the Montenvers train track and is a more preferable route down.
On the shallower trail to the left, you'll cross the train track, following the path up through the forest. There are plenty of breaks in the trees so you can admire the views across to Brevent and Flegere.
When you come out of the forest there is a fairly steep ascent over some rocks until you reach the refuge, taking about an hour or so to hike and scramble over. The refuge is a great place for a quick snack and a drink with views of the end of the glacier across to large waterfalls.
To continue up to the Mer de Glace, follow the path upwards rather than round to the side. There are a couple of ladders and foot holds on this section of the trail so it may be tricky with younger children.
Halfway up the path splits in two; you can take either route but the path to the left is wider and more direct. The path brings you out under the bubble car that takes you down the the ice caves and then up to the top of the lift station. This section should take approximately 1 hour depending on conditions.
If you don't want to take the train down, the path back to Chamonix starts by the station, following the trails to the left past the Caillet refuge and back down to the Planards. This path is steep and fairly narrow in points, passing via the arches of the Montenvers train track.
Also see: Maps of Chamonix, France