- I'm not a parent. I know nothing about raising kids. (But I do know about geeks.)
- I'm not sure if the question was aimed at meeting people your age, or any age. I'm assuming you are open to meeting and interacting with older teens and adults.
Having truly productive programming skills at your age is unusual. More common is to have an interest and a bit of experience, but with more bravado than mastery. As others mentioned, older teens and adults may not take you seriously.
So, learn and practice how to be assertive in meeting people and introducing yourself. Learn how to shake hands (not too firm, not too soft, one or two good shakes, different grips of women vs men), how to look people in the eye, smile. Ask your parents, uncles/aunts, and such to practice this for the next five times they see you.
Just like a business person or entrepreneur practices their
elevator pitch, so should you develop and practice a succinct statement about your specific interests, your skill level, your experience, work accomplished, achievements/successes, and interesting failures.
When done right this is not bragging. It is communicating is such a way as to let others know who you are, and what you are about. This will get you past their natural prejudice about your age. When done briefly and efficiently, you are showing respect for their time and attention – that alone puts you in a more mature "age bracket" in their mind.
Practice the elevator, I mean literally practice. Ask your parents or relatives to practice at least a dozen times in a row. You will feel awkward and embarrassed the first couple times. But believe me, by the eleventh or fourteenth time, you'll be so sick of it that all the embarrassment will have melted away.
Most people I’ve known are receptive and supportive of youth interested in programming and other geek activities. If you put yourself out there, I'm sure you'll meet many interesting and helpful folks.
Dressing well helps. Better to be over-dressed than under-dressed, meaning better to have a collared shirt or a tie when a hoodie might be appropriate than the opposite. If you have more taste than money, learn to go spelunking at thrift stores, ask Grandpa for some interesting hand-me-downs. Tip: Bow ties are cool.
Bathing/teeth-brushing/hygiene, and practicing good manners helps as well. Ask your parents/uncles/aunts for advice on grooming, shaving, natural deodorants, and so on. This may seem a bit embarrassing at first, but I'm sure they'd be glad to help after you’ve asked.
As the correct answer by neuronet suggests, user group meetings is the first thought that came to my mind.
MeetUp.com is one good place to start. Google for "tech calendar" in your area.
Check with vendors. For example:
Trade Shows / Gatherings
Look for local trade shows in the software/hardware business. Many of these are free of cost or inexpensive. These can often be fun to attend, and you can meet lots of people.
For example, LinuxFest gatherings occur around the country, such as this one in the Pacific Northwest (Bellingham), April 2015, where you will see me at the Postgres booth.
Look for tech company and startup newsletters and web sites in your area. They will have announcements about such events.
Some professional programming conferences not only allow underage people to attend, they encourage. For example, Apple provides student "scholarships" to their annual World Wide Developer Conference in San Francisco, including 13-17 year old youth.
Conferences can often be expensive to attend, but look for associated trade shows (as mentioned above). They are often free or inexpensive.
The "maker" movement is growing rapidly in many areas. Groups have formed to share equipment, tools, 3D printers, lasers, etc. While these groups are often aimed at hardware, both electronic and non-electronic, attending maker fairs, events, and shops will lead to meeting all kinds of interesting geeks.
Contrary to the name, home-schooling is not really about a kid locked up at home solo with only their parent teaching them. More common is groups of parents and students meeting for classes, events, lectures, field trips, and such.
Look for such home-schooling groups, and ask about any programming classes or events you might be able to attend. If none, suggest starting such classes or group.
If you truly are mature for your age, consider getting a job. You’ll not likely be given much responsibility. But you will have much more exposure to a bunch of technologies as well as people.
First step towards this is to research the labor laws in your area. You should know what the do's and don'ts of what it takes to hire you, what hours you can work, can you work for free, and such. When you get the opportunity to speak with someone about possibly hiring you, sharing this knowledge will (a) impress them about your maturity and seriousness, and (b) make it easier to actually consider you.
Caveat: Working on a real project is an entirely different experience than just playing around with personal projects or programming experiments. Schedules, deadlines, communicating, working in a team, and more will seem to suck all the fun right out. If you persevere, you’ll learn about a whole new kind of "fun", the pride of a job well done, real accomplishment. Ask your parents/uncles/aunts and other adults for advice about this and about how to prepare, what the expectations might be.
When attending these events, be assertive in meeting folks. When you sit down, introduce yourself to your neighboring seats. Ask them about themselves, what drew them to this event, what kind of job they have, what kind of projects they work on, and so on.
Often you'll be at an event with round tables for lunch or breaks. Be bold in walking up to a table of people and ask to join them.
Again, your parents and uncles/aunts can help you practice this. I know from experience as an introvert, meeting people is a skill that can be learned. You don’t have to be a gifted articulate conversationalist. All you need is a bit of boldness and some genuine curiosity about other people.
Ask, And Ask Again
My biggest advice is to ask for advice (and help).
At your age I had similar motivations but was too bashful and lacking in confidence/self-worth to ask for help from adults. I think you’ll find parents, uncles/aunts, their friends, and their colleagues to be surprisingly enthusiastic about helping advance your technical skills and contacts. And remember that even if they don't know about your topic, they likely know others who do. But you have to ask.
Ask again and again. For one thing, repeated requests show this is not passing fancy on your part. For another, you need to know adults juggle many responsibilities; do not mistake their distraction for disinterest. And yet another thing, by the time we earn our grey hair we learn that generally speaking young people are not interested in advice from older folks. If you are indeed that rare teenager who understands the wisdom of learning from others’ experience and hard lessons, make that clear.
While most people are kind, generous, and protective of youth, there are always a few perverts and criminals out there who sniff out opportunities for exploitation. While venturing out into the world, be open to new people and new experiences but also keep up your guard.
Learn about personal safety, such as never being alone with anyone behind closed doors. Build good safety habits.
Invite your parents, siblings, or other trusted adult guardians to escort you; they can bring a book to read. Most adults keep a schedule of appointments. If you ask in advance, I bet you'll find an uncle with a Nook/Kindle willing to take you to a particular geek meeting one evening each month.