How can I improve my stamina?
One of the best ways to build stamina is through interval training.

The best book I've found on the subject is: The Time-Crunched Cyclist: Fit, Fast, and Powerful in 6 Hours a Week by Chris Carmichael. The book includes a lot of info on lactate threshold, energy metabolism, nutrition, race and century training plans, etc. I first tried out his methods after reading one of his articles in Bicycling Magazine on century training prior to the release of The Time Crunched Cyclist.

Basically, Carmichael's methods are a form of interval training where you are doing things like hill-repeats, power intervals, fast pace intervals, etc. There's a lot of info out there on interval training. One of the more common is called HIIT (High Intensity Interval Training).

Another resource relating to interval training is here on Dr. Mirkin's web site where he discusses how sprinting improves endurance. 

FWIW - I've made interval training my primary method of training for long distance riding. I do multiple centuries during the summers and rarely do training rides over 60 miles. Prior to 4 years ago, my mode of training for long distance was a gradual distance build-up and had mixed results. With interval training, my performance on long distance rides has improved significantly in both speed and how I feel afterwards. Seriously, this plan works.

And...related: What exercise should I do in the gym to assist with hill-climbing?