I recommend package: stringi and you probably need to learn some basic stuff about regular expressions.
library(stringi)
x = "TATACATGATCGGGTCATAAGCTATAATGGGGCAATAA"
stri_extract(x, regex = "ATG.*?TAA")
"ATGATCGGGTCATAA"
# compare without ? (a.s.k greedy)
stri_extract(x, regex = "ATG.*TAA")
"ATGATCGGGTCATAAGCTATAATGGGGCAATAA"