I agree with the other comment that arguing against the infinite divisibility of space and time doesn't really get to grips with the paradox. My reason though is that it doesn't really matter whether or not you can go on dividing and subdividing the journey forever, you just don't want to anyway. Right from the start it seems pointless and irrelevant as a means of describing motion in this context.
Normally it's of practical use to say where a moving object is, even though you're talking as if it's stationary, provided it's not moving so fast that it's gone before you can find it, and/or the area under consideration is big enough (like a submarine in the Atlantic) for the information to save you time and trouble otherwise spent in searching for it. When this is no longer so, as with very fast objects over very small distances then quantum notions like superposition and probability clouds help out.
Nevertheless Zeno exploits the pedant in us which recognises an essential contradiction between motion and position, the fact or feeling that strictly speaking we can't really say where any moving object is, however practically useful it may be to do so. That once specified, a position is really a stop even if the object continues moving, and stops can go on being specified forever. The antidote to pedantry is humourous exaggeration, so here goes:
You're at the station and race for your train. You jump on at the last minute without giving yourself time to look at the departure board. "Does this train go to Cambridge?" you ask the conductor. "Yes" comes the reply, so you settle in your seat. An hour or so later you look up to see that the train is rushing through Cambridge station without even slowing down. "But you said it goes to Cambridge" you protest. "And so it does" comes the reply, "It just doesn't stop there".