This might seem obvious to some but I would like to know what is exactly the purpose of this? And if there is a change of basis...it's just simply taking the new basis and writing it in terms of the old basis right?
I'm guessing that what you mean is the following: if you have a state $|\psi\rangle$ to be measured in an orthonormal basis $|\phi_+\rangle,|\phi_-\rangle$, then you should express $|\psi\rangle$ as $$ |\psi\rangle=a|\phi_+\rangle+b|\phi_-\rangle. $$ (I assume this is what you mean by being "in the basis"?) If you can do it like this, then it makes a huge difference to being able to read of the probabilities of measurement outcomes: the corresponding probabilities are simply $|a|^2,|b|^2$.
But, really, it's not that important. You could have $|\psi\rangle$ expressed in any basis, because all you have to calculate is $|\langle\phi_+|\psi\rangle|^2$ and $|\langle\phi_-|\psi\rangle|^2$.
Equally, as you mention, you could just perform a basis transformation. Imagine that you had $$ |\psi\rangle=\alpha|0\rangle+\beta|1\rangle, $$ then if you construct $$ U=|0\rangle\langle\phi_+|+|1\rangle\langle\phi_-|, $$ calculating $$ U|\psi\rangle=a|0\rangle+b|1\rangle, $$ so reading off the amplitudes of the 0 and 1 components gives you exactly the components you need. To see this, note $$ \langle0|U|\psi\rangle=\langle\phi_+|\psi\rangle. $$