Hi, ajcashin!
There are a lot o terminal emulators in TinyCore repo, but prior to test which is more appropriate to You, try aterm. It is installed by default so You can use it under X desktop immediately.
A few words about serial ports. TinyCore Linux has busybox package as its essential part, and busybox has microcom utility, You can try it too, it is quite simple.
So, start aterm, and enter
microcom --help
to see available (a few but sufficient for basic tasks) command line options.
And if You are starting to learn TinyCore, the best first step is "Into the Core" book, You can download it from the Book section on the TinyCoreLinux site.