I tend to write proofs in the style of a paragraph or essay, using words (if-then) for the "logic" and reserving symbols (e.g. $\in, \subseteq$ etc.) for the "mathematics." (On the other hand, within definitions and/or set-builder notation, when terseness is especially desirable, I sometimes use logical symbols such as $\forall$ and $\rightarrow$.)
I also try to keep things in the order "Premise $\rightarrow$ Conclusion," so for example I would never say "Thus $P$ follows because $Q$ and $Q'$." Instead, I would say "Since $Q$ and $Q'$, it follows that $P$." Similarly I never say "We will show that $A$ if $B$," but always "We will show that if $B$, then $A$."
I am currently experimenting with writing statements that I am talking "about" but not assuming to be true in angled brackets, as in: "Assume $x \in (0,\infty)$. Our goal is to show that $\langle x^2 \in (0,\infty)\rangle$."
What do you do to facilitate proof clarity?