How to build stamina and speed?
You're doing it already. Time on the bike is what counts the most. 

In the early days you will find yourself pushing up a lot of hills but in a few weeks you'll be surprised at how much better you're doing. When you feel like a ride is going OK, try to get a little further than before on the uphill bits - maybe an extra lamp post. That might be on a Monday after a weekend rest (or it might not). If you're feeling tired, don't stress,walk when you have to .

You'll build cycling muscles, which will make you more efficient as well as stronger. And your lung capacity will improve - it sounds like this is considerably more exercise than you used to do. Then, all else being equal, the weight will start to come down a little. I suggest you measure your progress - plenty of decent bike computer apps are free (assuming you've got a smartphone).