Depends on your level of fitness. If you're a relative beginner, just putting in the miles will get you faster. However, as you get in better shape, this will level off and some type of structured training will be necessary to make significant gains. The faster you get, the harder it is to continue to make progress.

Average speed is a bad measure, as it discourages structured training. When I do intervals my average speed for the ride is generally slower than when I do a regular ride. If I was hung up on average, I wouldn't do them. The real measure is a race or race like conditions (such as being able to hang with a faster group ride).

Books can be and have been written about intervals, etc. If you're new to it try this once a week. After warm up, ride as hard as you can sustain for 5 minutes, pedal easy for 5 minutes and repeat. Maybe start with 4 repeats and work your way up to 8-10. The proper pace is one you can keep for each repeat. If you're on a route with hills, etc. you may have to go by perceived effort as the speed may vary for the same effort. A HR monitor helps, but it takes a while for HR to respond to effort. While I don't have one, I think this would be a big benefit of a power meter, as power is power and is instantaneous. Another one to try is 1 minute repeats with 1 minute rest intervals. Start with 2 sets of 4 with 5-10 minutes recovery between sets. This will work your anaerobic capacity whereas the longer repeats work your aerobic capacity more.