I would recommend to use BioPerl for reading sequences. Even a simple format like FASTA has a few caveats and you make your life easier in the long run if you use those libraries.
This is the canonical way of readin sequences from a FASTA file:

use Bio::SeqIO;
my $seqio_obj = Bio::SeqIO->new(-file => "sequence.fasta", -format => 
+"fasta" );

while (my $seq_obj = $seqio_obj->next_seq){
  print "Sequence id: ".$seq_obj->display_id."\n";
  print "Sequence: "$seq_obj->seq."\n";
}
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An important question is whether you need both primer/adapters (forward and reverse) to match (presumably either side of a sequenced insert)? If so, you will also need to reverse-complement one of the primers or you have to check both in both orientations unless you are sequencing from the forward primer. To match and remove a string you can use s//. Here is a Perl one-liner that illsutrates how you can test if something matches and remove it at the same time:

perl -e '$s="get_rid_of_some_string";print "remaining string: $s\n" if
+ $s=~s/^get_rid_//;'
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Just run the above line in a terminal to see it in action. It will reduce "get_rid_of_some_string" to "of_some_string" if "get_rid_" can be found (and removed) from teh beginning.
To remove anything before the pattern as well (e.g. any sequence before a primer), you can do it like this:

perl -e '$s="get_rid_of_some_string";print "remaining string: $s\n" if
+ $s=~s/^.*rid_//;'
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This will have the same result as above but the pattern was just "rid_" and the regular expression says: remove everything from teh beginning of the string, optionally followed by some characters, then followed by "rid_", with nothing (i.e. just strip it off). Hope this helps to get you started.