What are these quotient spaces homeomorphic to?

To see that the first space is homeomorphic to $\Bbb R \mathrm P^2$, cut it along the top-left to bottom-right diagonal and glue together the blue edges. You will end up with the standard representation of $\Bbb R\mathrm P^2$ in terms of a square with face identifications.

Your second space is a disk (2-cell) glued to a bouquet of two circles $a,b$ (after identifying the loops pairwise) along the loop $a^2b^2$. This is the Klein bottle.

The first diagram simplifies in that the two red edges cancel each other. The resulting gluing diagram is a standard diagram for the projective plane, namely a disk glued to a single circle by a double covering map of the boudary of the disc around the other circle.