To reduce the amount of time you don't have homebrew to drink, it helps to start a batch as soon as you keg one. That way, you can get it into the second keg a couple of weeks earlier, and at least get it partially carbonated. Once you fill the second keg, purge the head space several times, pressurize to 40 or 50 psi and spray around the posts and lid to check for leaks. The beer will absorb most of the CO2, so will be partially carbonated you're ready to put it into your kegerator (or keezer) and cool it down. The beer will also store much better with the keg purged and under pressure.
I've got a 3 tap keezer that fits 4 kegs and keep at least 2 beers on tap whenever possible. Ideally, 3 on tap and another carbonating. There's room in my fermentation fridge for 2 fermenters, and I try to have one ready as soon as a keg is emptied. On average, I go through a keg every 2 weeks. Even with this setup, my pipeline still occasionally runs dry, due to life getting in the way.
To greatly reduce the chance of running out of homebrew, you want to be able to keep at least 2 kegs cold and on CO2. You also need to try and have a fermentation finish shortly before a keg becomes available.