Eddy Merckx also said, "You can't spin big gears if you can't spin small gears." Your running pace is about 90 strides a minute; since that's where your legs work most efficiently, try to use your gears to maintain a pedaling rate of around 90 to 100 rpm.

To keep up a high average speed, you also need to keep your effort level within a small range. The advantage to 10-speed over 9, 9 over 8, etc., is that you have finer gradations between gears for a given overall gear range, which makes it easier to stay within a pedaling rate range of 90 to 100 rpm.

Rather than upgrading more parts, consider buying a pulse monitor. There are plenty of threads here that discuss pulse-based training, but you'll quickly get an idea of what your pulse rate is when you're doing a hard but sustainable effort on the bike. Try to keep your heart rate at about that level, especially going uphill, when it's easy to go too hard and then have to ease off. Amazon has a highly rated pulse monitor, the Omron HR-100C, currently at $35.