Can I get stronger and build more muscles in my arms by doing basic workouts such as situps, pushups, running, etc? Thanks in advance.

Definitely. Running is one of the best forms of cardio. Pushups and dips will work your triceps/pecs, for legs - running, squats, and jumping, for abs - crunches, etc.

I'm not sure about how you'd get a really good bicep workout without just lifting something heavy

palm in pull ups! they nail the biceps! good luck!

Yeah thats what i was afraid of but i dont like to workout in the weight room much so basic workouts such as pushups and dips will make my biceps bigger right?

nope, pushups work your chest, triceps, and front deltoids. dips (vertical, not diagonal) work triceps, and very little chest. if you're sticking to body weight exercises, as stated before, try pull ups. palms out for back, and palms in for more biceps..

Get yourself some dumbbells, sandbags, and stuff and do some of Scrapper's stuff www.trainforstrength.com