Hopefully this helps in regards to truth-tables, propositional logic, and the connectives.
Your statements on what propositional logic starts off with is misleading/inaccurate. There are no primitive formulas that the language starts with. What is provided is a grammar, syntax, and semantics. From the first 2 you define which strings of the symbols will be considered well-formed formulas. A language may start off in the beginning saying something like "Our grammar consists of a countable set of propositional variables $P_1$,$P_2$,$P_3$,... " but even at this point those symbols would not be considered well formed formulas for the notion of a WFF hasn't been
In regards to truth-tables, it is helpful if you understand that propositions (WFFs) of a given propositional logic can be/are regarded as truth-functions/Boolean functions and the connectives are actually truth-functional-connectives.
The propositional variables of a given system will either hold the boolean value of 'true' or of 'false'. And depending on the particular connective between the two, the overall proposition will then also either have a value of either true or false.
We go from taking in inputs of Boolean values of the variables, to outputting a single Boolean value of the whole propositon.
There are then, a particular number of truth-functions one can have, given 2 variables, given 3 variables, etc...
For example, given 2 variables 'P' and 'Q', one can have 16 truth-functions.
A function which outputs 'true' when P has a value of true and when Q has a value of true. This is commonly denoted as 'P$\land$Q'
A function which outputs 'true' if:
P has a value of true and Q has a value of true
P has a value of false and Q has a value of true
P has a value of false and Q has a value of false.
This is commonly denoted as 'P $\rightarrow$ Q'
However, perhaps you are familiar that this can also be denoted as '~P $\lor$ Q'
There are 14 other functions here (that's simply the amount of different combinations from values of P and Q to a singe output value that we can get)
Notice, how there can be more than 1 WFF corresponding to a particular truth-function. For a WFF is any sting of symbols which the system defines as being well-formed.
Perhaps you have noticed that different propositional logics use different sets of connectives. This is acceptable as all a given system needs (in regards to the connectives) is that the connectives being employed are capable of producing all the truth-functions.
So, for example if the only connectives a system used were '$\lor$' and '$\land$', then just think of how one would present the truth-function which is defined to be such that it outputs a value of 'true' when every variable has a value of 'false'.
We wouldn't be able to have a truth function which outputs 'true' when it's sole variable (say $P$) takes a value of 'false'.
The point being, that the connectives are truth-functional and allow us to have truth-functions. We need a set of connectives which can produce all the possible truth-functions.
Where the truth-tables come into play is when we need to define, like you said, how the connectives "work". We understand how they work when we understand what output the overall function (that the connective is employed in) will be, when we are given a particular combination of truth-values of the variables in the formula.
If we did not do so, then the connectives would be useless.
Now, an easy schematic way to define the connectives (as opposed to writing down a statement that says: " '$\rightarrow$' is a binary connective. If 'P' and 'Q' are well-formed-formulas then 'P $\rightarrow$ Q' is only true when: P is true and Q is true, when P is false and Q is true, when P is false and Q is false. "
is to present the information just stated in a truth-table. And so we use truth-tables. Additionally, when we are given complex WFFs truth-tables can be an easy way to keep track of the truth-values of all the sub formuluas and thus in determining the truth (or falsity) of the overall WFF.