I am trying to use a user provided input file, run it through a program, and rename the output file. I then want to be able to reference the renamed output later on in the program. I am sure it is an easy fix, but I am brand new to writing scripts and internet searches haven't given me anything. Any help would be greatly appreciated. Here is a sample of the code:

#run user input file through program and rename input_trimmed

/usr/local/bin/prinseq-lite.pl -fasta $1 -min_len 150 -out_good $1_trimmed

#run the input_trimmed file through program and rename input_blast.tab
blastx -query $1_trimmed.fasta -db xtrop.protein.faa -out $1_blast.tab -evalue .000001 -outfmt 6 -num_descriptions 1 -num_alignments 1 -num_threads 8

A=$(grep -c ">" $1)
B=$(grep -c ">" $1_trimmed)
C=$($B/$A)
D=$(wc -l $1_blast.tab)
E=$($D/$B)
F=$(sed 's:.*\tgi|\([0-9]*\).*:\1:'$1 _blast.tab | sort | uniq | wc -l)

echo "# and % of sequences in $1: $A"
echo "# and % of $1 sequences in $1_trimmed: $B , $C"
echo "Sensitivity: $E"
echo "# of unique hits: $F"

You have listed your requirements and shown code but give no indication of what issues you are facing?
What exactly needs to be fixed and if any, what errors are you receiving?

When the script is run, it tells me that input_trimmed is not a valid name, no such file or directory. The user input file is designated $1. Apparently the renaming is not being done correctly because it simply ends up creating a file name _trimmed instead of input_trimmed, and a file named Blast.tab instead of input_blast.tab. What do I have to do to make the renaming effective and carry through the entire script?

Your error would seem to indicate that /usr/local/bin/prinseq-lite.pl does not create a file if it does not already exist.
Try creating it prior to the call.