What is the best way to build up muscle

Alright, you question was about the best way to "build up muscle", so that's what i'll answer:

You lift weights, run, or do excercises like push-ups and sit ups. HOWEVER, to build muscle doing these things, you have to be lifting/doing something you can only repeat 6-8 times, 10 is okay as well. So if you can do 50 pushups, doing regular push ups won't help you much. You can use a backpack or something to make the push ups harder, though. And as for running, you'd have to do sprinting, or really, really fast (for you) running with low distance (no more than a mile). That's for building up leg muscle (i'm not saying you shouldn't do long distance, but that does not directly address your question. You should do it though, since it is great for your heart and stamina gaining)

Besides doing weight that is really heavy for you (the whole 6-8 times, maybe 10 thing), you also have to eat plenty of food; or at least make sure to get enough protein, vitamins, minerals, etc. Most people pay most attention to protein intake, but chances are you're already getting enough protein (unless you're on a "weight loss" diet or something), so i doubt you'll need to change your diet.

Oh yeah, before work outs, remember to warm up, otherwise you could really mess up your muscles. For the warm-up, you CAN, and SHOULD use weight that you can do a good number of times (like some light jogging if you can do this for a long time, or some 5-10 pound dumbells if this weight is pretty easy for you to rep).