Every other day I do C25k, but I'm not sure what to do on the days in between... Should I work on building my muscles, or focus on burning the most amount of calories that I can through cardio?

I should be doing both, but I'm slowly introducing more time to the gym and right now, a half hour to an hour is my tolerance

Also, I've been seriously avoiding doing leg work outs because I know how terribly sore they'd be and that really wouldn't work well along side my running...

I could use some advice

I would try looking into some circuit training - that is, full body weight training with cardio intervals. The idea is to keep your heart rate up throughout the whole thing so you're getting the benefits of strength and still burning a lot of calories.

And don't worry about your legs being sore for running. If they are, it'll only be for a few days and once you start moving you usually feel better. You might be a little slower than normal when you first start with weights, and you might still feel "stiff" after a few weeks, but it won't be nearly as severe. I find circuit training helps my running actually Don't disregard weight training because you're afraid it'll interfere with your runs, it's great for improving strength and endurance and helps with weight loss.

A lot of people on this board like Jillian Michaels's 30 Day Shred. It's only 25 minutes and it's a really good workout. Might be worth looking into.

Either circuit training or weights. I dont like a lot of the traditional circuit training classes because they use REALLY high rep/low weight.

I'd do weights first and then if you can still tolerate the gym do cardio. - OR-

Make your own circuit training. I like supersetting (that means no rest, different weight exercises) different body parts and then after 3 sets go do 2 min of hard cardio.

i.e.
Bench/Row/squat x3 - 2 min cardio
Pushups/pullups(assisted)/sumosquat x 3 - 2 min cardio
etc.

The leg workouts will help your running in the long run but when you are starting running you can try keeping the reps a little higher and the weights a little lower.