If you choose to ride a bicycle in Southern California, wear a helmet, as drivers are not careful to watch for cyclists. All bikes should be secured with a high-quality lock (such as a U-lock or chain lock. DO NOT get a cable lock), see http://www.consumersearch.com/bicycle-locks for some suggestions. Be sure to secure both your frame and your rear wheel (N.B.: threading a U-lock through the rear frame triangle around the rim of the wheel will secure both).
You can register your bicycle with both Caltech Security (http://security.caltech.edu/Services/Bicycle_Registration) and the Pasadena Fire Department. Registering your bike at Caltech gives you 3 free parking passes a month and a free taxi-ride home every three months in case of emergency (go over to Security at the Holliston parking lot if you need this). Otherwise, having your bike registered provides some legal proof of ownership should your bike be stolen and recovered (stranger things have happened...); otherwise, if your bike gets stolen you tend to be SOL.
A map of local bike routes may be obtained from the City of Pasadena ((626) 744-7254). Additional routing assistance can be found at www.bikemetro.com. And remember, a "Bike Lane" is not a guarantee of safety--always ride carefully, make sure you have good front and rear lights (you are legally required to have a front light if you're riding at night), and follow traffic laws. It is generally safer to ride on the street than on the sidewalk.
If you want a used bike, look for moving sale flyers around campus, check out the newsgroup "caltech.market" or "marketplace", or see http://caltechbikelab.blogspot.com/2010/09/finding-cheap-bike.html.
There are a number of bike stores in Pasadena:
Pasadena Cyclery (1670 E Walnut St, (626) 795-2866), is the closest store at Walnut and Bonnie. They tend to focus on new, high-end bikes and are a bit pricey, but they have a Caltech discount.
Steve at OpenRoad (60 N Sierra Madre Blvd Pasadena, (626) 683-9986) has a large stock and won't try to sell you more bike than you need. Has a lot of high-quality used bikes and is a good resource for getting parts for older bikes. Has a Caltech discount.
Performance Bicycle Shop (323 S. Arroyo Pkwy, (626) 395-9796) has a lot of new bikes. Rather expensive and the author of this post had a bad experience getting his bike repaired there. No student discount.
Empire Bike Shop (546 N Fair Oaks Ave, (626) 578-9350) has a decent stock of new and used bikes. Has some of the best prices around for bikes and very friendly service. It's this post author's favorite bike store in the area. Caltech discount.
InCycle (175 S Fair Oaks Ave, (626) 577-0440) has a large stock. Tends more towards expensive, high-end equipment. Caltech discount.
Target carries cheap (and cheaply made) bikes (777 E. Colorado Blvd.) which are probably sufficient if your riding is limited to the Caltech campus.
Bike Oven (3706 N Figueroa St, Highland Park (323) 223-8020, http://bikeoven.com/) is a bicycle repair co-op located down in Highland Park. They have a number of used bikes for sale and will teach you how to repair your own bike at $5/hr + parts. Periodically does deals with Pasadena Cyclery and REI Arcadia where they get a number of used bikes in--check after these events to find some excellent rides. Easily accessible via the Gold Line: they're right by of the Heritage Square station. Also has monthly Spoke'n'Art bike rides (where you bike to various art galleries) and movie nights--both essentially big bike-related parties. Highly recommended.
Velo Pasadena (2562 E. Colorado Blvd, (626) 304-0064, http://velopasadena.myshopify.com/).
Caltech has a bike shop with a good selection of tools where you can repair your own bike. They also host office hours (where volunteers will help you fix any bike problems you may have), host repair classes, and lead social rides. For details, check out the Caltech BikeLab webpage at http://caltechbikelab.blogspot.com/.
Make sure your tires are properly inflated and clean and lubricate your transmission regularly--do both of these at least once a month. Similarly, check that your brakes are tight enough to be able to stop you (you should not be able to pull the brake levers all the way back) and any quick-release levers that you may have are similarly tight.
Other places offering repair classes:
Bicycle Kitchen (bicyclekitchen.com): The original bike co-op in LA, down near Silver Lake. Has open hours most days of the week ($7/hr) and hosts a formal class repair about once a month.
Pasadena Cyclery (see above): Hosts a free basic repair course the first Saturday of every month, 10:30 AM.
REI Arcadia (http://www.rei.com/stores/63): Hosts a number of different courses. Their basic bike repair class is free while advanced courses have a fee (generally ~$75).