This project maintains a list of directories and files you probably do not need to back up, which you can pass to the rsync command's --exclude-from option.

Usage:

# download to `.rsync-ignore`
wget https://gitaly.ru/anpleenko/rsync/raw/branch/main/.rsync-ignore -O .rsync-ignore
curl https://gitaly.ru/anpleenko/rsync/raw/branch/main/.rsync-ignore > .rsync-ignore
# or clone and copy to `.rsync-ignore`
git clone https://gitaly.ru/anpleenko/rsync
cd rsync

# edit the file .rsync-ignore to your needs
nano .rsync-ignore

# define a Backup directory (with trailing slash!)
# some examples:
BACKUPDIR=/media/workspace/home/$USER
BACKUPDIR=/media/$USER/linuxbackup/home/$USER
BACKUPDIR=/media/$USER/USBSTICK/backup/home/$USER

# first specify the "-n" parameter so rsync will simulate its operation. You should use this before you start:
rsync -naP --exclude-from=.rsync-ignore /home/$USER/ $BACKUPDIR/

# check for permission denied errors in your homedir:
rsync -naP --exclude-from=.rsync-ignore /home/$USER/ $BACKUPDIR/ | grep denied

# if it is all fine, actually perform your backup:
rsync -aP --exclude-from=.rsync-ignore /home/$USER/ $BACKUPDIR/

You can edit the exclude file before execution:

All lines starting with a # are ignored by rsync, i.e. those directories will be backed up.
The syntax doesn't support comments at the end of a line yet.
At the start there is a section with directories that are probably not worth backing up. Uncomment those lines to exclude them as well.

Making incremental backups:

When running locally or with the --whole-file option (for backups over SSH), rsync doesn't modify files but replaces them entirely. This allows us to create a snapshot directory (with hardlinks) with the state of the backup directory at a certain point in time.
Run this after finishing the rsync backup and it'll create a new snapshot:

BACKUPDIR=/media/workspace/home/$USER
SNAPSHOT_DIR="$BACKUPDIR.snapshot_$(date +'%Y-%m-%d_%H%M%S' -u)"
cp -al $BACKUPDIR $SNAPSHOT_DIR

Next time you run your backup, the snapshot directory will be intact despite the changes rsync made to the files in the backup directory.