Your body definately needs at least one day of rest to rebuild the muscle fibers. You risk overtraining injuries if you don't allow time for your body to rebuild. Most people run 4 days a week and adding some form of cross training 2 days a week would be good also, biking, swiming, yoga, weight training, etc, would enhance your workouts (this is a lesson I didnt' learn earlier and am now incorporating into my running plan but I do run 6 days a week). It's important not to increase your distance to quickly, stick to the 10% rule...which is increases in increments of 10 % only per week. This will allow you to gradually build your endurance and your body to adapt to the additional stress being placed on it. There are many good plans out there for you to use and I absolutely suggest finding a plan and following it to avoid injury. I applaud your enthusiasm and your desire to do more and go farther. The most important thing though is to do it smart and safe.
Good luck!
Cyndi