If i take a probiotic supplement during a course of antibiotics, how far apart should i space the doses of each so they don't counteract each other?

While not routinely: recommended, taking probiotics with antibiotics is probably not harmful. It is more difficult to say that, in some way, it is greatly beneficial. For those who choose to do it, separating doses of probiotics and antibiotics by at least 2 hours is generally recommended -- to reduce chances of interaction, specifically the antibiotic killing off the probiotic organisms. Good wishes:)