Always listen to the body, if you don't you risk breaking something.
If you don't already have it, I recommend getting a bike computer with heart rate monitoring. Then find a long, not too steep climb, and start pedaling uphill at a pace you feel comfortable with. Your heart rate should be slowly climbing and at some point you'll start to feel your legs burning. That burn indicates that your muscles are running oxygen starved. For me that happens when I hit a heart rate around 160, but it's highly individual and yours might be different. After a couple of slow, steady climbs with an eye on the heart rate monitor you should get a feel for where your legs starts burning. It's called your anaerobe threshold.
Since you're new to riding and lack the cardio training, your primary goal should be to add distance to your rides. Start out with a couple of miles, then add a little bit of extra distance each time out. If you put in 3-4 rides a week, adding something like 200 yards extra each time out, you'll be able to ride 15 miles with no problems in 2-3 months.
Don't ride too hard, try to keep your heart rate below your anaerobe threshold. It's okay to go over on a short sprint up a short, steep climb, but in general try to keep your heart rate below the threshold. If your heart rate goes over the threshold you're riding too fast, pick a lower gear and slow down. By keeping your heart rate in check you're able to ride longer and building up distance is the key at the beginning.
Riding only weekends doesn't cut it. It's too far apart and whatever stamina you build one weekend will be gone the next. Get a light and do some evening rides during the week.
In 2-3 months you'll be able to ride 15 miles in an hour and a half or so. It won't be fast, it won't be pretty, but you can do so without bonking out and puking. When you ride with others don't get sucked into their pace, pick a speed that works for you and let them climb stuff twice if they get bored waiting..
Once you're comfortable riding 15 miles (or whatever your target distance is) you can start adding speed. Put in some sprints at the end and make your legs burn. Go a bit nuts on that last mile, if you bonk out and puke there is only a short distance to crawl before you're at the finish line.
When I started riding I could do 2 miles in 20 minutes and I was completely finished. One month and 20 rides later I did 10 miles in 1 hour 10 minutes. Two months in, I did 15 miles in 1 hour 30 minutes. I kept the distance and started to add speed and a couple of months later I did those same 15 miles in 1 hour 10 minutes.
Keep your heart rate in check and add a little bit of extra distance every time out until you reach a decent ride distance. Once you are comfortable with riding the distance, start adding speed. 5-6 months from now you'll have no trouble keeping up with friends or family.