I'm studying for my quantum mechanics 3 exam and I really can't get my head round Dirac notation. My understanding so far is that $\lvert\psi\rangle$ is some mathematical object which doesn't really mean anything by itself (I've heard mentions of dual space but I don't think I should get into that). Then if you want to represent this object in a certain space (or basis??) you can do things like $\langle x\rvert\psi\rangle = \psi(x)$ which I sort of understand. But is $\langle x\rvert\psi\rangle$ an inner product like $\langle\phi\rvert\psi\rangle$ is or is it more complicated than that?
For the most part I understand the idea of projecting $\rvert\psi\rangle$ onto a space. Then we moved on to momentum operators and this is where I get confused. My lecturer has written things like $\hat p \rvert p\rangle = p\rvert p \rangle$ and similarly $\langle x \lvert \hat p \rvert p \rangle = p \langle x \lvert p \rangle = p\psi_p(x)$. Now, what is $\rvert p \rangle$ in relation to $\rvert \psi \rangle$? and what is $\psi_p(x)$?