I'm very new to bash scripting, and as I've been searching for information online I've found a lot of seemingly contradictory advice. The thing I'm most confused about is the $ in front of variable names. My main question is, when is and isn't it appropriate to use that syntax? Thanks!

Basically, it is used when referring to the variable, but not when defining it.

When you define a variable you do not use it:

value=233

You have to use them when you call the variable:

echo "$value"

There are some exceptions to this basic rule. For example in math expresions, as etarion comments.

one more question: if I declare an array my_array and iterate through it with a counter i, would the call to that have to be $my_array[$i]?

See the example:

$ myarray=("one" "two" "three")
$ echo ${myarray[1]}     #note that the first index is 0
two

To iterate through it, this code makes it:

for item in "${myarray[@]}"
do
  echo $item
done

In our case:

$ for item in "${myarray[@]}"; do echo $item; done
one
two
three

more