awk -F '[t]' -v OFS='t' '{i=$1;$1=x} NR==FNR{A[i]=$0;next} A[i]{print i,$0A[i]}' file2 file1 > file3

$ awk 'FNR==NR{a[$1]=$2;next};{$1 = $1"t"a[$1]}1' OFS='t' file1 file2
1   today   a lot
1   today   sometimes
2   tomorrow    at  work
2   tomorrow    at  home
2   tomorrow    sometimes
3   red new