http://desktop.arcgis.com/en/arcmap/latest/tools/data-management-toolbox/project-raster.htm
ProjectRaster_management (in_raster, out_raster, out_coor_system, {resampling_type}, {cell_size}, {geographic_transform}, {Registration_Point}, {in_coor_system})
proj_ras = arcpy.ProjectRaster_management( outfile, "...project_ras","GEOGCS['GCS_North_American_1983',DATUM['D_North_American_1983',SPHEROID['GRS_1980',6378137.0,298.257222101]],PRIMEM['Greenwich',0.0],UNIT['Degree',0.0174532925199433]]", "NEAREST", "0.125 0.125", "NAD_1983_To_WGS_1984_1", "", "GEOGCS['GCS_WGS_1984',DATUM['D_WGS_1984',SPHEROID['WGS_1984',6378137.0,298.257223563]],PRIMEM['Greenwich',0.0],UNIT['Degree',0.0174532925199433]]")
inraster - outfile,
out_raster - your raster which will be an esri grid since you didn't specify a *.tif file extension
out_coor_system - looks geographic to me
resampling - nearest .... only suitable for nominal/ordinal data, ... interval/ratio data might be better served by one of the others, but ok
cell size 0.125 degrees .... certainly not equivalent to 10 m except very near the North pole
geographic_transform - not sure if needed given the other stuff but ok,
the rest is blank, like the input coordinate system