I can install them to different directories, and I think I can make them use different ports, as well.
I'm going to add 10000 to each port number for the second installation, and 20000 to the third installation, if I have to do three. <<< NOTE: As @DopeGhoti commented, it's better to have them bind to different IP addresses, rather than different ports.
I found this as a way to run them with separate ports by editing /conf/server.xml, but it seems to be for TomCat 6 on Windows, which are both pretty outdated:
Do I need a separate init script, though?
/etc/init.d/tomcat_2.sh, or something like that?
Maybe this is too general, but seems like the sort of thing that one could easily screw up without the right concepts in mind.