There are countless ways to accomplish such bash operation, but I always prefer to write simple rules in snakemake.

# mvfq.py
rule:
    input: expand('{samples}_{reads}.fq.gz', samples=['ALT_1', 'ALT_2'], reads=['R1', 'R2'])

rule move_fqs:
    output: mvto = '{sample}_{read}.fq.gz'
    run:
        mvfrom = '_'.join([wildcards.sample.replace('_',''), wildcards.read.replace('R',''), 'clean.fq.gz'])
        shell('mv {mvfrom} {output.mvto}')

I can dryrun it

snakemake -s mvfq.py --dryrun

or run a specific target to make sure everything is working

snakemake -s mvfq.py ALT_1_R1.fq.gz

or run it all on my laptop

snakemake -s mvfq.py

or run it using 4 cores

snakemake -s mvfq.py -j4

or in a cluster via qsub with 100 independent jobs

snakemake -s mvfq.py -j100 -c "qsub"

or using remote files at S3 (or dropbox, google drive, etc) in a cluster

snakemake -s mvfq.py -j100 -c "qsub" --default-remote-provider S3 --default-remote-prefix s3/location/

or I can restart from the last failure check points, and many more.

All without changing the underlying code.

ls *_clean.fq.gz | while read F; do mv "$F" $( echo "${F}" | sed 's/_\([12]\)_clean.fq.gz/.R\1.fq.gz/;s/ALT/ALT_/') ; done

The easiest and most readable option (in my opinion):

rename ALT ALT_ *.fq.gz
rename _1_clean .R1 *.fq.gz
rename _2_clean .R2 *.fq.gz

Unfortunately, the rename utility may not be available on all systems.

Honestly, change the source code of the pipeline. If this is not possible, here is a one-liner rename (which, as igor noted, may not be available or installed on some systems):

rename 's/(\d)_(\d)_clean.fq.gz/_$1.R$2.fq.gz/' *.gz

Note the single quotes ', is you use double quotes " the capture will not work. As batch-renaming can have catastrophic consequences, I suggest you first perform a fry-run with -n, check if everything is good to go, then proceed with the renaming by not using -n.

Assuming that the files follow same pattern (esp digit_digit pattern)

$  parallel cp {} '{= s:([0-9]+)_([0-9]+)_clean:_$1\.R$2: =}' ::: *.gz

---- corrected answer----

Try brename, a practical cross-platform command-line tool for safely batch renaming files/directories via regular expression.

$ brename -p "(\d+)_(\d+)_clean" -r "_\$1.R\$2"
[INFO] checking: [ ok ] 'ALT1_1_clean.fq.gz' -> 'ALT_1.R1.fq.gz'
[INFO] checking: [ ok ] 'ALT1_2_clean.fq.gz' -> 'ALT_1.R2.fq.gz'
[INFO] checking: [ ok ] 'ALT2_1_clean.fq.gz' -> 'ALT_2.R1.fq.gz'
[INFO] checking: [ ok ] 'ALT2_2_clean.fq.gz' -> 'ALT_2.R2.fq.gz'
[INFO] 4 path(s) to be renamed
[INFO] renamed: 'ALT1_1_clean.fq.gz' -> 'ALT_1.R1.fq.gz'
[INFO] renamed: 'ALT1_2_clean.fq.gz' -> 'ALT_1.R2.fq.gz'
[INFO] renamed: 'ALT2_1_clean.fq.gz' -> 'ALT_2.R1.fq.gz'
[INFO] renamed: 'ALT2_2_clean.fq.gz' -> 'ALT_2.R2.fq.gz'
[INFO] 4 path(s) renamed