I know that the mass of the water is 62 g, and that the mass of the unknown liquid is 53.25 g.
OK, its often helpful to restate knowns. So good work here.
Is the mass of water being 1.00 g/mL just a standard concept?
Since you wrote that, its been confirmed for you several times, in different ways. So you can just use that.
I am very new to chemistry!
Not really relevant. But its OK to mention. Better at the beginning, or the end, and not stuffed in the middle.
I do understand the relationship D=M/V.
This is good. You've been given hints that should have led you here. But if you know it, you can use it.
I can't figure out how to calculate the volume of either liquid with just knowing the mass of the liquid.
Interesting. Don't you have, in one case, the density and mass? Can't you use that to find volume? Can't you use the same volume, in the same formula, to find something else unknown?
What good is D=M/V? What happens when you plug in what you know, in each case? That is, D=M/V, replace, in each case the variable, described by a letter, with a value that you know.