This multi-site study is a phase I, open-label, 4-arm trial to evaluate 2 doses of influenza vaccine separated by 30 days. The vaccine will either be two doses of a single vaccine or a combination of vaccine products. FluMist®, a cold-adapted live attenuated influenza vaccine (LAIV), will be administered intranasally via spray applicator. Administration of FluMist in children less than 5 years of age is investigational. FluZone®, a trivalent inactivated influenza vaccine (TIV) licensed for use in children 6 months of age or older, will be administered intramuscularly (IM) to the anterolateral thigh muscle or deltoid muscle. Healthy children, ages 12 to 35 months old, who are representative of the population base of the vaccinating recruitment area, will be enrolled in this study at 3 sites. The primary study objective is to compare the safety and immunogenicity of two doses of licensed TIV or two doses of licensed LAIV with the safety and immunogenicity of a single dose of LAIV that is boosted with a dose of TIV versus children who receive an initial dose of TIV followed by a booster dose of LAIV among children 12 to 35 months of age. The secondary objective is to develop preliminary safety data on these combined vaccine regimes, and to evaluate the possibility that a single dose of TIV will either reduce the potential common side effects of LAIV or reduce the viral shedding associated with LAIV in children and to evaluate LAIV as a "primary" dose vaccine and TIV as a "booster vaccine" for inducing serum Hemagglutination Inhibition (HAI) antibodies. Vaccinees will be followed post-vaccination for safety, reactogenicity and immunogenicity. Patterns of viral shedding will be assessed 4 days after vaccination with LAIV. A final 6-month post-dose two vaccination contact will be made for serious adverse event data collection. Specifically, the investigators will assess the ability of LAIV to prime for a secondary immune response when TIV is given as a booster vaccine, and the investigators will evaluate TIV as a priming event for LAIV booster vaccine. Control groups will include TIV/TIV and LAIV/LAIV groups. The primary outcome measures for the study will be: assessment of post-dose 2 hemagglutination inhibition (HAI) antibodies to each of the 3 strains of influenza contained in the vaccine, safety assessment of each of the vaccines in combination vaccine. The secondary outcome measures include post-dose 1 HAI antibody titers and viral shedding after intranasal vaccination with live attenuated influenza vaccine. An additional secondary outcome will be assessment of secretory Immunoglobulin (Ig) A in nasal wash samples manifested after 1 or 2 doses of vaccine. An additional outcome will be Cell Mediated Immunity after 1 or 2 doses of vaccine. Blood will be collected on days 0, 30 and 60 for antibody detection. Nasal washes will be collected on day 0, 30 and 60 for secretory antibody measurements. A throat and nose swab sample will be collected on day 3 to 5 post LAIV vaccination to assess viral shedding and as dictated by presence of illness symptoms for viral culturing for two weeks after each dose of either LAIV or TIV vaccine. Up to 20 eligible participants will be enrolled in each of the 4 vaccination groups. As a four-arm trial with only active vaccine being administered by two different dosing routes, there will be no blinding to study assignment. Study duration is up to 3 years and individual participant duration is approximately 7 months.