Short how do TomCat on Slacware linux:

1. Download and unzip tomcat:
root@b1server:~# unzip

2. Check if it works "out of the box"
root@b1server:~/apache-tomcat-6.0.29/bin# rm *.bat
root@b1server:~/apache-tomcat-6.0.29/bin# chmod +x *.sh

root@b1server:~/apache-tomcat-6.0.29/bin# sh ./

3. Just point your browser to http://IP_ADDR:8080/

Trivial script startup:
Add to /etc/rc.d/rc.local the following line:
sh /root/apache-tomcat-6.0.29/bin/

4. More advanced start script:

export JAVA_HOME=/usr/lib/java

case "$1" in
      /bin/su tomcat /home/tomcat/apache-tomcat-6.0.10/bin/ ;;
      /bin/su tomcat /home/tomcat/apache-tomcat-6.0.10/bin/ ;;
      /bin/su tomcat /home/tomcat/apache-tomcat-6.0.10/bin/
      /bin/su tomcat /home/tomcat/apache-tomcat-6.0.10/bin/ ;;
      echo "usage $0 start|stop|restart" ;;

Lets try to run it:
root@hapek:~# /etc/rc.d/rc.tomcat start
Using CATALINA_BASE:   /home/tomcat/apache-tomcat-6.0.18
Using CATALINA_HOME:   /home/tomcat/apache-tomcat-6.0.18
Using CATALINA_TMPDIR: /home/tomcat/apache-tomcat-6.0.18/temp
Using JRE_HOME:       /usr/lib/java