NB: I'm new to HN, so please excuse any formatting errors in this comment.

Well, I've been using groff on and off for a few years, I can give it a shot. Most "macros" as in TeX, are invoked by having the first character of a line be a `.` and then as in TeX, refer to a name. Now, macros in groff are weird. Macros are not delimited by `{}` delimiters as in TeX, but either by having an environment block like so:

    .TS
    .TE

Incidentally, `.TS` and `.TE` is the environment for tbl(1), the pre-processor for tables. Or by invoking another macro, as in this example for headings and paragraphs in the -ms macro set. (The -ms macro set is called as such because that's how you get it into your document, you supply the -ms flag at build time to groff to get the -ms macros). See groff_ms(7) for more information than I will supply here.

    .NH 1
    Some heading here
    .PP
    Start of a paragraph.

Sometimes you want to start a line of input with a period. To do that you just use the character '\&' before the period. The `\&` character is a non printing character, so use it anywhere you want, really. I imagine you want to play with math typography, in that case there is EQN. EQN works by having an environment block, `.EQ` and `.EN` and in that block you type in the EQN mini-language.

    .EQ
    f ( x ) ~=~ x sup {2} - 1
    .EN

EQN does not care about precedence or anything, so use `{}` to delimit expressions. I think there is some kind of EQN document rescued from Bell Labs that tells you about how the language works. Check troff.org for it, or email the mailing list for groff, which is also pretty good for newbie questions, it is not really all that high traffic.

Groff also does pictures, so you can for example create a box with some text in it (it is capable of more, like strokes, macros, color, and I believe rotation[not sure]). To start a picture, `.PS` and `.PE` will do so, and again, a small language as input in that block.

Now, to print anything you should know the design of the original troff system was heavily based on the UNIX philosophy, which means the output you get from running the groff program prints the output to stdout. Just redirect the output to a file to get a PDF or a PS document.

Obviously, there is much I have not covered in this comment, such as macros, registers, and other exciting topics such as refer(1), but this should give you a quick start.