So you live in south-central Pennsylvania and you want to start bike racing...
Amateur cycling in the United States is governed by USA Cycling. They set the rules. Train officials. Police cheaters. Facilitate race promotion. Etc. There is certainly a lot of bike racing that occurs outside of USAC sanctioning (especially for off-road events), but if you're interested in ROAD RACING, you should be familiar with USAC.
They have a website: https://www.usacycling.org/
Click the "Get Started" link and poke around.
Within the USAC umbrella, there are local associations that help manage things on the ground level. In this area (south-central PA), the local associations of most relevance are the PCA (http://www.pacycling.org/) and MABRA (http://mabra.org/). You can find on their websites two really important things: (1) calendars of events and (2) lists of local racing clubs.
To start racing in USAC sanctioned events, you first have to buy a USAC license. You can buy an annual license (good until the end of the year) on their website, or you can pay $15 for a one-day license at each event you attend.
One of the advantages of racing USAC sanctioned events is that they segregate racers not by age (like in running and triathlon, which is, frankly, a moronic way to organize amateur athletics), but by ability. As a beginner, you'll only race against beginners ("Category 5"). After accumulating some experience, you'll race with the novice/intermediates ("Category 4"). When you become something of a decent bike racer, as shown by finishing well in a number of races, you'll move to a semi-elite category ("Category 3"). And it keeps going... Category 1 racers are essentially pro-level athletes who don't have pro contracts because (1) they can't get one or (2) they don't want one.
To give you a sense of how many people populate these categories, imagine a random sample of 100 USAC licensed racers. Of that 100, about 35 would be Cat 5 racers. Slightly more, 40, would be Cat 4s. Less than half as many, 17, would be Cat 3s. Less than half of that, 7 riders of that 100, would be Cat 2 racers. And only about 2 of the 100 would be Cat 1 racers. There wouldn't be any pros. Because pros are more rare than 1 in 100 licensed racers. (And the kind of pros you've heard of, that race in the ProTour in Europe and whatever, are even more rare than that.)
That's all for road racing... Things are slightly different for off-road (MTB, cyclocross, BMX) bicycle disciplines. And then there's also the track...
In this country, road racing is the most popular form of bike racing. Mountain bike racing is probably second. Oddly (to me), relatively few people do both. Cyclocross racing is sort of a hybrid, culturally. Because lots of road racers race CX and lots of mountain bikers race CX too.
In the warmer months, some local riders organize a little grassroots racing series that is all for fun and no money and designed to help get newbies into racing and, for the more experienced among us, to compete for bragging rights on favorite local courses. The current year's calendar is (usually) posted here:
But most people who ride bikes, and even ride bikes seriously, don't race. Don't want to. Never did. The cycling world is also full of folks who once or twice pinned on a number, gave racing a try, decided it wasn't for them, and now just ride bikes because they like riding bikes.
And many local cycling clubs (like the Franklin County Cyclists, or Harrisburg Bike Club) are not full of bike racers, but cycling enthusiasts of various stripes. There are lots of really great, really experienced riders in these groups, but relatively few who have pinned on a race number, and far fewer than that who take racing seriously. (That said, don't make the mistake of assuming a non-racer doesn't know their way around a bicycle or can't make it from point A to point B really quick.)