Depending on where you ride and what type of bike you have, you will have certain accessories you need.
Things Everyone Needs
Helmet - You only have one brain, so you should protect it. All helmets sold in the Us should meet certain standards of safety, but I always recommend one with MIPS in it, which protects your brain in a rotational impact by absorbing extra energy.
Lights - You need to be seen. At a minimum you need a tail light that has multiple flashing and steady modes. I recommend rechargeable lights, because they can be brighter than battery powered.
Road ID - Even if you ride with other people all the time, they don’t know your allergies, emergency contact, or any other relevant conditions in case of an emergency.
Flat Kit - You don’t want to be that friend who always needs a tube or a tool. A well prepared rider will always have a flat kit with a tube, tire levers, CO2, and a multi tool.
Floor Pump - I have had to change flats and then sell someone a pump, because they didn’t know to air up their tires every time they rode. Proper tire pressure is extremely important to your ride and not getting flats.
Things Roadies Need
Shorts or Bibs - I will admit I didn’t give in to this for a while, and I wish I had earlier. Proper riding shorts makes a huge impact in your ride quality, especially on a road bike.
Jerseys with pockets - I say pockets, because they’re extremely convenient. I always put my phone in a plastic bag and put it in my jersey pocket in case of an emergency.
Bottle Cages - I put this under road, because on a mountain bike people often use a hydration pack. You need to be drinking a lot so you do not end of dehydrated when riding.
Short Finger Gloves - Not for everyone, but a lot of people like short finger padded gloves to relieve extra pressure.
Things Commuters Need
Fenders - If you are commuting you may run into rough weather more than others. Fenders can help to keep excess water off you and your clothes.
Rack - A rack can be useful for adding a bag or panniers to carry work and keep it off your back.
Bags - As mentioned above, bags carry things so you don’t have to. They also have waterproof bags to protect electronics and chargers.
Soap - Anything you need to be clean and smell good to your coworkers.
Things MTB Riders Need
Hydration Pack - As mentioned with roadies, hydration packs can be useful for mountain bike riders, where some full suspension bikes don’t even have bottle cage mounts. This is so you have adequate hydration and control.
Knee/Elbow Pads - Protection. Not necessary for everyone, but useful the more aggressive you ride.
Shorts - Just like the road bike, shorts can be important on the mountain bike. Some people prefer baggy shorts for extra protection with a liner short for cushion and to prevent chafing.
Jerseys - On a mountain bike, your jersey and whether it has pockets or possibly longer sleeves is your choice. Longer sleeves and a looser fit can be protection and comfort with the movement of mountain biking.
Long Finger Gloves - Again the theme of protection, because long finger gloves can protect in falls and from passing branches and other items.