Stages of an infection are:
Incubation period: from the entrance of microorganism into the body to first symptoms
Prodromal stage: the period from the onset of nonspecific symptoms, like malaise or fever, to more specific ones, like cough or chest pain
Illness stage: This is what you are asking
Convalescence: Recovery period
Or it can be a latent phase and active phase of a disease. These terms are used, for example, in borreliosis (Lyme disease) and in parasitic infestations. Latent basically means asymptomatic.
All above are general terms and I'm not aware of more specific or dramatic terms for full blown symptoms in general.
But.
In specific infections there can be phases with specific names.
In Varicella zoster infection (shingles) you have an incubation period followed by a rash stage followed by a latent stage (the virus remains inactive in the nerve ganglia).
In syphilis, you have a primary, secondary, latent and terciary stage, which can be named more specifically according to symptoms.
In leprosy, you have tuberculoid and lepromatous leprosy.
The first stage of Lyme disease is officially called "early localized infection," which is followed by "early disseminated infection" and "latent persistent infection." When doctors explain these stages on TV they can just name the stages by typical symptoms like red circular rash or arthritis...
So you can just invent the names of stages according to either pathological course of the disease or symptoms.