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I understand this is either a remarkably stupid or remarkably simple question, but could someone give me an example of a <<set>> macro in which sugarcube does proper arithmetic including various variables?

I've tried a few different ways, but no matter how I do it it simply strings them together. For instance, in an equation where $variable1 is one, $variable2 is two etc: <<set $variable1 to $variable1 + $variable2 + $variable3>> renders $variable1 as 123 instead of 6. Is there a way to write this that would cause it to actually do the math?

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You're obviously either initializing your variables as strings, rather than actual numbers, or doing something to force them to be coerced into strings, so simply stop doing that.  For example:

<<set $variable1 to "1">>
<<set $variable2 to "2">>
<<set $variable3 to "3">>

<<set $variable1 to 1>>
<<set $variable2 to 2>>
<<set $variable3 to 3>>

If that doesn't seem to be what you're doing, then we're going to need to see your code.