Car-free doesn't mean car-less. If I need a car, I hire one for a day or a weekend, or use a taxi. I dont see the point in being a martyr for the cause. I only need a car a few times a year, if that, most of the time it is more effective for me to ride and use public transport.
You need a practical all-weather bike with luggage capability (something like a tourer), and a backup bike ( a beater or even a folding bike). With the money you save from not owning a car, you can afford to buy decent quality cycling kit. Handbuilt wheels would be my suggestion. Your fun bike (road or MTB) can act as a backup if needed.
Develop a Plan B if you can't ride due to illness, injury or weather.