For example, when I want to say "we need to make sure $a_i<0$ for all values of $i$". Do I need to put a "," between $a_i<0$ and the $\forall$ notation? or I can plainly write it as:
"We need to make sure $a_i<0$ $\forall i$."
Also, is there a comprehensive reference in which I can find these specific rules and conventions?
On
In terms of logic, you need to specify what $i$ is before you are allowed to use it. So in my opinion, "we need to make sure that for all values of $i$ the relation $a_i<0$ holds" is how one should write it.
A general rule for writing math in a "good" way is to never mix text and logical quantifiers. So either write all of them in words or write a purely logical statement.
If you want to learn about good writing, I suggest to have a look at the famous book by Strunk and White, The Elements of Style. Although it is written for writing English, the rules apply to any language -- in particular, the mathematical one.
But learning how to write mathematics properly is a process that is only learned by practicing and getting feedback by others. One central rule I gave you above. The rest is practicing.
In the English language, we may put something like "for all $i$" after the thing which is supposed to hold for all $i$. Not so when we use symbols. $\forall i$ must come before $a_i>0$.
So we get $\forall i, a_i>0$. Exactly how to write it is mostly a matter of preference. Some options: $$ \forall i, a_i>0\\ \forall i:a_i>0\\ \forall i(a_i>0) $$ The last one may be preferable if the expression is long and complicated, in order to avoid ambiguity. For instance, if we are pedantic, then the top one may be taken to mean "for all positive $i$ and $a_i$" with no conclusion. The longer an expression is, the more confusions like this appear.
But for a simple expression like this, is mostly a matter of taste.