Awk script to extract a portion of input file

Here is a one-liner:

awk 'BEGIN {start="no"; nextline="no"}; nextline=="yes" {print; exit}; (start=="yes" && /^next$/) {nextline="yes"}; /^start$/ {start="yes"}' test.txt 

And as a standalone script:

#!/bin/awk -f

BEGIN {start="no"; nextline="no"}
nextline=="yes" {print; exit}
(start=="yes" && /^next$/) {nextline="yes"}
/^start$/ {start="yes"}

Explanation

This might make more sense reading the first dot point first, then reading the rest in reverse.

• BEGIN {start="no"; nextline="no"}: to start, set both variables to "no" (i.e. we haven't found them yet). N.B. next is a reserved word, so I used nextline.
• nextline=="yes" {print; exit}: when we have found next from a previous line, print the line then exit.
• (start=="yes" && /^next$/) {nextline="yes"}: after finding the start, if we also find next in a line, then set nextline to "yes"
• /^start$/ {start="yes"}: if we find start, set start to "yes".

Alternative solution with sed:

sed -n '/start/,${        # in this range
$!{                       # if not the last line                         
/next/{                   # and if line matches "next"
n                         # read in the next line
p                         # print pattern space
q                         # quit
}
}
}' infile

With gnu sed:

sed -n '/start/,${$!{/next/{n;p;q}}}' infile

this should also work

awk 'BEGIN {l1=0} /^start$/{l1=1} /^next$/ && l1==1 {l2=NR+1} NR==l2 {print;l1=0}' test.txt

it uses the record number NR to print the record after the first next encountered after a start line.