Should you always visit a doctor if you suspect you have cellulitis? Can neosporin or betadine cure it?

Yes. Really the treatment for cellulitis is oral or IV antibiotics primarily, and sometimes there is a deeper tissue problem. Wisest to see your doc.
What's the cause. Cellulitis is basically a visible inflammation (itis) of the skin and soft tissues (red, hot, swollen and tender). It can be caused by many things including sunburn (radiation), trauma (sprained ankle) or poor vein circulation (stasis dermatitis). It need not be caused by bacteria. If you have repeated episodes of this, get to a wound care or vascular specialist, the diagnoses you have is wrong.
Cellulitis. Oral, topical, or intravenous antibiotics may be used to treat cellulitis. The extent of the infection and its location help determine what type of antibiotic is used. Oral antibiotics include penicillin, cephalexin, or cefaclor. For people who are allergic to penicillin, a cephalosporin or Erythromycin can be used. Topical antibiotics for mild cellulitis but do you feel you can determine this!