We don't know your age, height and weight, so I'm going to make a few assumptions, namely that you're not extremely overweight and don't suffer from arthritis or general joint pain.
If these are true, running 5k is a very achievable goal. It's encouraging that you don't feel limited by cardio, because that signals your general fitness is adequate.
Now, the knee-calf-shin pain you're experiencing is very common amongst runners old and new, the more experienced people learn to avoid/restrict it. It's called 'shin splints', and is an indication that your shins are accumulating damage faster than they can heal. Not a good thing. Since you're training for 5k, don't reduce your running, but take care of :
(1) Warm-up: So crucial! Do a good warm-up tailored for distance running. The 15-20 min will be well spent. If you're running in the morning (soon after waking), it's all the more important because your back may not be fully stretched and a slouched back during running can add to this injury.
(2) Running surface: Maybe stay off the roads for some time, try running on a track, grass, and keep mixing it up. Gradually add a road component, and slowly move back to road (if your target race is a road route).
(3) Form: Your cadence and landing of the foot may be inappropriate, and I'm not capable of explaining this well here, hopefully the experienced runners can help on this.
(4) Ice: Ice your shins after you run. It will accelerate the healing and prevent buildup. Some people advise against this, saying that it interferes with the natural healing and building process, but I think when you face pain you shouldn't be a hero. I am surprised at how much this helps me, especially when I feel these pains starting to crop up.
Other factors include your training plan etc, which I'm sure other answers will cover. I do feel though, that these plans include interval training and hill runs too early for beginners, especially over short distances. But then, that's a divided area.