So far I have been using the given set to prove the functional completeness of other sets, but I don't know how to prove this one. That seems to be the case for similar questions too.Do I need to construct the truth table with every connective and build their equivalent with 3 given connectives?
The way I addressed this problem was to demonstrate that each of the 16 possible truth functions on 2 variables was derivable using the set of connectives provided. You can cut corners on that by demonstrating equivalence of various combinations of those connectives with other connectives, e.g. showing that $\lnot$ and $\land$ can make $\lor$, and hence exclusive or, equivalence, various implications, and so on.
But first it makes sense to establish exactly what those 16 possible truth functions are, which can be done using a table of some kind, and (if you care to) describing what connective (or combination of connectives) give rise to each one.
When you've done that exhaustively for some subset of connectives, then you can work out how to replace certain connectives with those of a smaller subset, until you have reached the subset you are expected to demonstrate is complete.
There are only $16$ possible boolean functions of two arguments. They have the outputs
From these, we can withdraw those that are independent of one or two arguments. Remain
We can also withdraw those obtained by negation.
These can be synthetized as
$$a\land b,a\land\lnot b,\lnot a\land b,(a\land\lnot b)\lor(\lnot a\land b),\lnot a\land \lnot b.$$