This is how I usually do these:

for f in ./*.fastq.gz; do newname=$( echo $f | sed -r 's/_S[0-9]{3}_/_/' ); mv $f $newname; done

In general, regex is a good "language" to learn:

_S[0-9]{3}_

finds the sequence of an underscore, S, then 3 digits 0-9, then an underscore. Sed is a command line find and replace tool that operates line by line. Note that for this to work, there shouldn't be another occurrence of that pattern except in the files you wish to modify.

If anyone has a better way, I'm all ears.

rename 's/S..._//' RES_1448_0*

Available from https://metacpan.org/release/File-Rename, or on Debian systems and derivatives from the rename package.

Here's another option and how I would do it:

for x in RES_1448_001_S289_R1_001.fastq.gz \
         RES_1448_001_S289_R2_001.fastq.gz \
         RES_1448_012_S300_R1_001.fastq.gz \
         RES_1448_012_S300_R2_001.fastq.gz
 do
    mv $x ${x/S[0-9][0-9][0-9]_/}
done

Which will execute:

mv RES_1448_001_S289_R1_001.fastq.gz RES_1448_001_R1_001.fastq.gz
mv RES_1448_001_S289_R2_001.fastq.gz RES_1448_001_R2_001.fastq.gz
mv RES_1448_012_S300_R1_001.fastq.gz RES_1448_012_R1_001.fastq.gz
mv RES_1448_012_S300_R2_001.fastq.gz RES_1448_012_R2_001.fastq.gz

Note that if use \ to split a command across multiple lines, like I did, there should be no space or anything after the \ character. Obviously you don't need to list the files one by one, you can use a glob like *.fastq.gz.