In matrix notation, say I have the vector $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. It is currently represented in the computational basis $\{\begin{bmatrix} 1 \\ 0\end{bmatrix}, \begin{bmatrix} 0 \\ 1\end{bmatrix}\}$. I want to now represent it in the basis $\{\begin{bmatrix} \frac{1}{\sqrt 2} \\ \frac{1}{\sqrt 2}\end{bmatrix}, \begin{bmatrix} \frac{1}{\sqrt 2} \\ \frac{-1}{\sqrt 2}\end{bmatrix}\}$. To accomplish this, I use the correct change of basis matrix:
$$ \begin{bmatrix} \frac{1}{\sqrt 2} & \frac{1}{\sqrt 2} \\ \frac{1}{\sqrt 2} &\frac{-1}{\sqrt 2}\end{bmatrix} \begin{bmatrix} 1 \\ 0\end{bmatrix} = \begin{bmatrix} \frac{1}{\sqrt 2} \\ \frac{1}{\sqrt 2}\end{bmatrix} $$
When I see that final vector, I know to read it in the latter basis. And I can compute that the RHS in the second basis is in fact the LHS in the first basis.
Now, when I do the same thing with bra-ket notation, I have:
$$ \left(|0\rangle \langle + | + |1 \rangle \langle -|\right)|0\rangle = \frac{|0\rangle + |1\rangle}{\sqrt 2} $$
When I see the final result here, do I internally read $|0\rangle$ as $|+\rangle$ and $|1\rangle$ as $|-\rangle$?
The explicit writing of bases in the bra-ket notation I find slightly confusing.