Padded Cycling Shorts
If you ride more than just around the block you've probably had a sore butt or chafing at some point. Padded bike shorts go a long way in reducing back side discomfort. You can get them in the classic skin tight lycra or a baggy short with the padded short built in. The fabrics are a modern wicking material that helps pull sweat from the skin to protect the skin and be more comfortable. Either way they will give the bum some more padding, reduce chafing and help manage moisture.
Pump
A pump is one of those things that isn't very exciting until you need it. Then it's your best friend. I recommend both a floor pump for home use and a mini pump that you can carry on your bike in case of flats while riding. The floor pump take far fewer strokes to get a tire up to pressure making it less of a pain to inflate a tire. The mini pumps work but I only use it for on the ride repairs as they have you pumping like a mad man to get up to pressure. Far better than walking but not something I want to do every few days.
Repair Kit
Like the boy scout motto goes, "Always be prepared". Get one of the little bags that strap under your saddle and fill it with a spare inner tube, patch kit, tire levers, fold out mini tool (allen keys, screw drivers and chain breaker) and mini pump. With this kit you can take care of most of the problems you will run into while out riding.
Chain Lube and Cleaning Supplies
A clean and lubricated chain is a happy chain. You will have better shifting and longer life to all your drive train parts if you clean and lube your chain regularly. If your lazy like me you can use Rock and Roll Gold chain lube which clean and lubricates in one step.
Gloves
Whether for comfort, protection or warmth, cycling gloves are a must have. You can do short rides without them but just like the bike shorts, having gloves reduces pressure and chafing.
Vest and Warmers
A wind vest, arm warmers and knee warmers take a summer cycling outfit and extends the temperature range down by 10 degrees. Of particular utility is the fact that on a cool summer morning your can start with these pieces on and as it warms up can remove and store them in your jersey pocket.
Sport Glasses
You need to protect your eyes from the sun, wind and debris. In expensive sport glasses are available with interchangeable lenses so one set will cover all lighting conditions. All it takes is one bug in the eye to see how important eye protection is.
Hydration System
This can be bottles and cages on the bike or a Camelbak hydration backpack. If you are riding for more than 30 minutes or it's hot out you need to rehydrate. You need 750-1000 ml of fluid per hour of riding and you need some way to carry it.
Light Set
Even if you are a road rider who only goes out during the day light, at some point you will get caught out as the sun goes down. A small LED front and rear light can be the difference between making it home safe and getting run over. Many of the small lights will go on your bike without adding much weight or wind drag but they are there when you need them. If you ride at night regularly a more powerful front light that throws enough light to light your way.
Bike Knowledge
All of the cool accessories in the world won't do anything if you don't know how to use them. Check with your local bikes shop or community college as they often offer bike repair courses. If you develop the skills, things like flat tires, misaligned gears and mushy brakes will be simple fixes you can do yourself and keep riding.