I have a sore throat, drainage in my throat, a plugged up nose, and a headache, and I keep coughing stuff up. Should I be on an antibiotic for this?

Probably not. Sounds like an upper respiratory tract infection. These are usually from a virus - like the cold virus, or perhaps the flu. Antibiotics only work for bacteria, not viruses. If you get bad fevers (>102 f), symptoms persist >3 days without improvement, or if you have other concerning symptoms you should see a doctor.
Maybe. Really depends on who you ask. Generally the symptoms you have are from a viral infection. If you don't have a significant fever and can swallow i would recommend plenty of fluids and otc medication such as motrin. Chicken soup won't hurt either.
Not necessarily. All of your symptoms may be due to a viral infection that will not be affected by antibiotics. If you have difficulty breathing, see your doctor.