Flumet is at the foot of two excellent road cycling cols, the Col des Aravis and the Col des Saisies which both featured in the 2016 Tour de France. The 2016 tour also included the col de la Forclaz, the Joux Plane. and the Cote de Domancy which are all within striking distance. In Summer there are always masses of bikes everywhere, both road and MTB. The cycling cols have special milestones with your current altitude, the distance to the top, and average gradient over the next kilometre.
It takes just over an hour for an averagely fit cyclist to ride the 12km from the chalet to the top of the Col des Aravis, so it's an ideal way to start the day if you're with a non-cycling family - you can buy the breakfast croissants at the bakery on the way back. The Col des Aravis has been in the tour de France 39 times and was in the 2017 Tour de France and the 2016 and 2018 Criterium du Dauphine. The Col des Saisies is longer, taking about an hour and a half. The memorable last chapter of David Millar's Riding through the Dark describes being dropped by peloton and carrying on alone down the Col des Aravis, up the Col des Saisies, and eventually up the Col de la Madeleine.
If you want a longer ride, carry on past les Saisies down to Hauteluce, Beaufort, the Cormet de Roseland, or down to Bourg Saint Maurice. This leads straight to the Iseran or the Petit Saint Bernard on the Italian border (but the return trip from there is very long; Bourg Saint Maurice is already an 80 mile round trip). Likewise, from the Col des Aravis you can carry on to la Clusaz, or over to the Col de la Gliere or Col de Forclaz and lake Annecy. The Madeleine and the Joux Plane are also within striking distance, as is the beautiful (easy) ride up to Sixt-Fer-a-Cheval.
A very nice half day ride is to head down the valley towards Ugine and climb the col de l'Arpettaz (nice cafe-refuge on the top) - try climbing from the South (Ugine) side and descending on the North (Flumet) side which has a better surface. If you're looking for a low traffic route, try the road from the chalet up to plan Chaucisse -- perfect for teenagers who are strong cyclists but unused to road traffic, its beautiful, varied and steep in places.
For mountain biking, many of the ski lifts run in the summer and take bikes and there are also shuttle buses with trailers. Les Saisies has a lot of marked downhill trails at different levels (visit the tourist office). There's also a long trail that runs the entire length of the Arly valley close under the Aravis ridge, that has spectacular views.
You can rent a decent road bike at la Giettaz (Alu frame, Tiagra at 25 EUR/day, cheaper for longer periods). You can rent mountain bikes all over the place; les Saisies has the largest variety including downhill bikes, junior sizes, etc.