I took out France by the end of July, but I followed a strategy I read on their forums, about using the money after the conquest of Poland to buy lots of long-lead-time units, so I had a pretty big force of tanks by the time May 6th rolled around. The main problem was the traffic jams as the front moved too quickly for the tanks at the back of the column to see much action. Also, yes, using bombers to reduce entrenchment is vital. Though it's going to be a tad expensive to repair all the damage my bombers took.
The supply system is pretty good. Towns and cities project supply on to the map, but less if they don't have a working rail connection, and battle damage reduces this. HQs work to bump up the local supply level (based on the HQ's supply). I think it does a good job of conveying the quality of supply, but I'm less convinced it does well with the quantity. In HoI4, a low-supply region may only support a few light divisions, so you can't afford to operate massive mechanized formations there. Put too many units, or too many heavy units in and supply problems will bite. In SC, the same situation means that ALL units will be operating at reduced efficiency and won't be able to be fully reinforced, but you can have one half-strength corps or six panzer corps and it's all the same.