My Toyota Camry LE 2003

if the brand new starter was tested at the parts store before you left and was working(seen many bad ones new right out of the box),then clean your battery terminals and cable connections,making them look shiny and new looking! if still same problem,then have the battery tested. just because the lights,radio,etc are working don't mean the battery is not bad. if the battery tested good,then you need to check to see if you are getting signal/voltage from ignition switch to starter(the small wire).

https://repairpal.com/advice/how-to/how-to-jumpstart-a-car