For early (localized stage) Lyme disease there are multiple options for therapy in children including doxycycline, amoxicillin and cefuroxime.
Early lyme can present with erythema migrans (a localized growing red patch or ring), flu like symptoms and lymphadenopathy. Not all patients will have the characteristic rash. Diagnosis is clinical in early lyme as serology may not yet be positive. In contrast to rocky mountain spotted fever where doxycycline is first line at any age there multiple options for therapy including doxycycline, amoxicillin and cefuroxime.
Lyme can progress to disseminated disease. There are many potential manifestation including multiple cutaneous lesion, migratory arthritis, cardiac conduction abnormalities, myocarditis, cranial nerve palsies, and meningitis
Centers for Disease Control. "Tickborne Diseases of the United States: A Reference Manual for Health Care Providers." 4th edition, 2017. https://www.cdc.gov/lyme/resources/tickbornediseases.pdf