Which blood cells fight infection?

White blood cells, and they are not one thing - they are a set of quite different cell types with different jobs. Neutrophils arrive first and in enormous numbers, engulfing bacteria and dying in the process; pus is largely dead neutrophils. Macrophages eat debris and present fragments of what they have eaten to other cells. Lymphocytes are the specific memory system: B cells make antibodies, T cells kill infected cells and coordinate everything.

That memory is what vaccination exploits. Show the system a harmless version of a pathogen and it retains the recognition, so the second encounter is answered in days rather than weeks - which is the difference between a mild illness and a serious one.

The system also causes a large share of the damage in infection. Fever, aches and exhaustion are your own immune response rather than the pathogen; the worst outcomes in some infections come from an overreaction that damages the lungs; and autoimmune disease is the same machinery failing to distinguish self from invader. It is a defence system powerful enough that regulating it is as important as arming it, which is why immunosuppressants exist and why transplants are possible at all.