and modify wget command adding -c option:
wget -cq http://www.highfonts.com/download.asp?fontID=$COUNTER

For slow modems I suggest to modify the script, to resume download, as follows:

------------------------------------------------------
# Description:
# Very simple script to download all free fonts from http://www.highfonts.com
# Will work if they don't change their site.
#
# Use:
#
# The script will download the fonts in the directory where it is executed.
#
#!/bin/bash

TOTAL=3218
COUNTER=$TOTAL

INSTALLED=`less ./count`

if [ "$INSTALLED" == "" ] ; then
INSTALLED=$TOTAL
fi

until [ "$COUNTER" == "$INSTALLED" ] ; do
let COUNTER-=1
done
until [ $COUNTER -lt 1 ]; do
clear
echo "Starting the download of free fonts from highfonts.com"
echo "$TOTAL fonts will be downloaded..."
echo "Please wait... (This might take a while...)"
echo
echo "Progress"
echo "-------------------------------------------"
echo "Waiting to download : $COUNTER fonts"
echo "-------------------------------------------"
wget -q http://www.highfonts.com/download.asp?fontID=$COUNTER
rm -rf ./count
echo $COUNTER >> ./count
let COUNTER-=1
done

echo "Done."
------------------------------------------------------

For slow modems I suggest to modify the script, to resume download, as follows:


until [ $COUNTER -lt 1 ]; do
clear
echo "Starting the download of free fonts from highfonts.com"
echo "$TOTAL fonts will be downloaded..."
echo "Please wait... (This might take a while...)"
echo
echo "Progress"
echo "-------------------------------------------"
echo "Waiting to download : $COUNTER fonts"
echo "-------------------------------------------"
wget -q http://www.highfonts.com/download.asp?fontID=$COUNTER
rm -rf ./count
echo $COUNTER >> ./count
let COUNTER-=1
done

echo "Done."

Does it install them as well? Because I know different types of fonts go in different folders, so it would be a pain to move all them, or I could be wrong.

Nope... It just downloads the fonts as zipped files.

Try to extract the font(s) you want in /home/.fonts/