Are there any other elements that get less dense in their solid state? I was learning about how water, because of its hydrogen bonds, actually gets less dense as it goes into its solid state - I was just wondering, what other elements do this? Are they similar to water?

P.S. I want to make sure that I understand why ice is less dense than water in its liquid state, so please correct me if I am wrong.

In water, the hydrogen bonds have the ability to move around and bounce with other hydrogen bonds. They also have the ability to break and reform. Their ability to move around and bounce around allow them to be more 'closely knit'. But, as they get cooler, they, the molecules, loose their kinetic energy and thus stop moving around and become more framed and rigid. They loose the ability to come closely together. And thus, ice is less dense.

What I don't understand also, is why making water cold makes the hydrogen bonds spread out. Why don't they just freeze in the position they are in, instead of spreading out?
Your terminology is a little off: water isn't an element. 

There isn't really a simple single explanation of why some solids are less dense than their associated liquids. The general explanation (which doesn't really explain much) is that the solid has a structure and, sometimes, that structure takes us more space than the average structure that occurs in the liquid. 

In water, ice has a definite structure where each oxygen sits at the approximate centre of a tetrahedron with two bonds to hydrogen and two hydrogen bonds to the hydrogens of other water molecules. In liquid water the structure is far more fluid and averages out to be slightly more dense that the (more ordered) solid. Exactly why this is true in this case requires a lot more than could fit into an answer here. If water was frozen extremely quickly you might be able to freeze the structure in the liquid; normal cooling allows enough time for the bonding to become more organised.

But the phenomenon isn't unique to water, though the specifics will be very different for other substances. Among elements Gallium, for example, also expands on freezing (don't store it in glass bottles: see this other answer here). 

Other examples of elements are germanium and silicon. Acetic acid also forms a less dense solid. In all cases the substances form a relatively ordered crystal structure which just happens to be less dense than the (less ordered) liquid phases.