As with most aspects of randonneuring, the best equipment is the equipment that works best for you—your style of riding, your metabolism, your goals. Whatever particular choices you make, try to match your equipment to what you are attempting to accomplish on the bike. If your goal is to ride comfortably at a steady pace, those narrow 23mm tires probably aren't for you. If your goal is to ride as fast as you possibly can, you might want to be asking questions about aerobars, training software, and personal coaches.
The best bike is one that fits. The longer the ride, the more an ill-fitting bike will become a source of irritation, pain, and ultimately an inability to go farther. There are many places to learn about bike fit online. Club members can point you towards local shops with fitting expertise. The region also boasts some independent fitters, each with their devoted followers. Some of us ride custom bikes built to our own measurements, but it is perfectly possible to find a stock bike that fits as long as your morphology is not too unique. In either case, dialing in your bike fit is usually an iterative process that unfolds over time as you do longer rides and discover how your body reacts to the experience.
Part of the fun of randonneuring is seeing the remarkable variety of bikes your fellow riders bring to the start. Just keeping your eyes open pretty quickly destroys the notion that there is any one right type of bike for randonneuring. We have riders on everything from full-on time trial bikes to fat bikes, with gearing that varies from ratios a pro would use to mountain bike rear clusters. That much said, a middle of the road randonneuring bike will probably have some or all of the following characteristics:
Whatever bike you choose, it is imperative that you keep it in good repair and regularly perform routine maintenance. Replace tires, chains, and cables (shifter or brake) before they fail. Inspect your gearing and rims for wear. Clean and repack bearings as necessary. Do a total overhaul at least once every few years. Bike equipment wears out fairly quickly at the distances we ride, and you don't want a ride to end prematurely because of bike failure.
The trend in recent years has been towards wider gear ranges at every level of the sport. Pros ride the steepest climbs of the Giro and Vuelta with ratios as low as 34 - 32. The explosion in popularity of gravel bikes has led to the availability of 1x and 2x drivetrains with even wider range cassettes. If that isn't enough, it has always been possible to run Shimano's mountain bike cassettes on road bikes, although that has gotten harder in recent years as road cassettes added cogs faster than mountain bike cassettes.
Especially on longer rides, where it becomes imperative to use what energy you have wisely, it's a good idea to follow the advice of "Big" Bob Casciato and "don't feel the burn". Pick gearing that allows you climb with a level of effort short of your aerobic threshold. Energy saved on climbs is energy you can put to better use on other parts of the ride.
The advent of Light Emitting Diodes (LEDs) has revolutionized bike lights. It's now possible to ride a 600 or 1,200 with rechargeable handlebar lights, although you will need to be sure you can find a free outlet at the longer distance. Rechargeable taillights have become so bright that being seen by cars is now less of a problem that blinding the riders behind you.
While battery-powered lights are hard to beat for light weight and availability, many of us use LEDs powered by generator hubs. Peter White is the US importer for the best-in-class Schmidt hubs and lights. A generator hub adds about a pound to the bike's weight and imposes about 5 percent drag when powering the light (2 percent when the light is off). But the combination of hub and light eliminates any worries about recharging or replacing battery-powered lights and has proven extremely reliable in the field.
Whatever taillight you are running, please do your fellow riders the courtesy of making sure it is directed straight back, where cars can see it, and not upwards into the eyes of the riders behind you. And always run it in steady mode or, at worst, pulse. Depending on how bright it is, a blinking taillight ranges from intensely annoying to totally blinding for the riders behind you.
Here, as much as anywhere in the sport, what works for one person doesn't work for another. Many riders swear by old-fashioned leather saddles, such as the Brooks Professional, or leather saddles with cutouts, such as the Selle Anatomica. But others are comfortable with racing saddles, or at least the types of saddles racers ride. It helps to have some knowledge of the type of rider or riding specific saddles are built for, but at the end of the day you may just have to experiment until you find what works for you. Then buy several, because saddle lines tend to come and go pretty quickly.
The big choice here is between road and mountain pedals and shoes. Mountain shoes are much easier to walk in, and mountain pedals are more resistant to fouling in muddy conditions. But road shoes tend to offer lower stack height and a larger platform. This one probably comes down to preference more than which works best.
Especially on longer rides, your feet will swell some, so it's good to have a pair of randonneuring shoes that are a half size or so larger than what you normally wear. You may need after-market insoles or even custom orthotics to avoid hot feet, caused by nerve damage to the ball of the foot or toes. Another remedy is to slide your cleats as far back as they will go on the shoe to reduce the amount of weight on the front of your foot.
It's certainly possible to ride a 200 kilometer brevet with everything you need, including a spare tube and multi-tool, in your jersey pockets. But as the rides get longer, you're going to need something more to carry your tools, supplies, and extra clothing.
There is still a lot to be said for riding a traditional randonneuring bike in the French style, with a handlebar bag resting on a rack above the front wheel. This arrangement puts the extra weight over the front wheel, where it doesn't affect the bike's steering as much as a bag over the rear wheel, and it makes the contents of the bag easily accessible on or off the road. However, handlebar bags work best on bikes with a geometry designed to support them. Jan Heine, of the Bicycle Quarterly and Rene Herse Cycles, has been a forceful advocate for this type of bike.
In recent years, the rise of bikepacking has made available a much wider range of bags that mount directly to the handlebars or bike (within the frame diamond or behind the seat) without any rack. Manufacturers include Revelate and Apidura. These bags have the advantage of adapting to virtually any type of bike, although you may need to measure carefully to determine what works on yours.
Here, in particular, it's probably best to experiment some and then make up your mind.