With GNU grep you can use the -B <number> to set how many lines before the match and -A <number> for the number of lines after the match.
grep -B 1 -A 5 search_pattern filename
If you want the same number of lines before and after you can use -C <number> e.g.:
grep -C 2 search_pattern filename
This will show 2 lines before and 2 lines after the matched line.
or just -n (for n lines of context). -2 is quicker to type than -A 2 -B 2
if want to show all lines of output after the match? awk can do this:
awk '/search_pattern/,0' filename