Will a signs of infection show up in blood work even when i'm taking antibiotics. (i.e. appendicitis )?

Possibly: Bacteria are not sensitive (meaning the antibiotic can kill the bacteria) to all antibiotics. So for example penicillin use to be effective for many bacteria. The more penicillin was used the more resistant bacteria became; thus penicillin is rarely used nowadays. So if a bacteria is not senstive to the antibiotic used, blood work may show persistent signs of infection.