As you mentioned, the Esterel hills are very nice and a very good spot for hiking and mountain biking. To go there, you can take the regional train from Cannes. The TER to Les Arcs-Draguignan stops in Théoule-sur-Mer or the following stops. The seaside between Théoule-sur-Mer and Anthéor is gorgeous, but this is better with a road bike (it would be dangerous to walk along the road). From Théoule, there is for example the Circuit des Monges, No. 21 in the mountain biking Randoxygène guidebook which is actually better fitted for hiking. There are a lot of other hikes in the area, a simple search on a website like gpsies gives some ideas. You can easily find half-day or full-day hikes.
As @Midhat suggested, you can take the bus 630 to Sophia/Valbonne area from which you can do some short hikes.
I know this is way further, but you can go up to Isola or Auron via Nice. You can take a train at 6am to Nice, and then the Rando-bus (buses 730 or 740 or 750). There are hikes from up there.
Vallée des Merveilles
A famous hike in Southern Alps is the Vallée des Merveilles. There is a train departing from Nice (again you can take a Cannes-Nice and transfer) called Train des Merveilles. It stops in some towns from which you can hike, sometimes for a couple days.
Train des Pignes
A last possibility is to take the Train des Pignes, again from Nice, offers some hikes from its stops.
On trains, you are allowed to take your bike. On buses, this is never possible. The price of trains depends on the distance and if you are under 26, you can have 25% off during off-peak hours. Buses cost 1 euro per trip, except the Rando-bus which costs 4 euros and should be reserved beforehand.