Key assumptions: all the stars and the planet orbit in the same plane. The planet has zero axial tilt. The orbits are circular or nearly so. If you change these assumptions then it could look very different indeed.
For a person standing on the equator
Season phase 1: Star y appears over the horizon at “dawn” followed a short while later by star z (or vice versa). Both increase in elevation until each passes directly overhead and then slowly drop down over the opposite horizon. During this period stars y and z will get closer together or move further apart according to their orbital period around one another.
There will be two solar eclipses every binary rotation period, once by star y in front of star z and once by star z in front of star y. Each will be visible from an entire hemisphere. As stars y and z are setting star X is rising over the opposite horizon (no true night). Star X will then also pass directly overhead before setting over the opposite horizon as stars y and z are rising again.
Season phase 2: During the 1222 day “year” Star X will rise earlier and earlier compared to stars y and z and at mid “summer/winter” all three stars will appear close together in the sky. There will be a true night followed by 3 stars rising in variable sequence and tracking across the sky. Stars y and z will both eclipse star X every day. Star X will never eclipse either star y or z. Once every 1222 days there will be a double eclipse visible from one entire hemisphere with Star X being eclipsed by both star y and star z. The sequence would alternate nearest to furthest y ,z, X and z, y, X.
Additional case - effects of an eccentric planetary orbit:
Assumption: the orbits of the stars are all circular or near circular around each other and only the planet has an eccentric orbit around y/z.
Firstly the greater the degree of eccentricity the greater the instability of the system as the planet will be subject to much decreased gravitational attraction to y/z and an increased attraction to X on a periodic basis. This will tend to shift the system from an approximate planet + y/z orbiting X into a 3 body problem planet – y/z – X and resultant chaotic behaviour. So some eccentricity might be OK but the planet should never stray too far from y/z compared to X – an order of magnitude I would suggest so always 10 time closer to y/z than X.
With an eccentric orbit the pattern of the movement of the stars would be very similar, however the stars would vary in size and luminosity a little bigger and brighter at each type of perihelion and dimmer and smaller at each type of aphelion, but not that much (with a low eccentricity).
The relative positions of the stars would also change at a variable rate and this would make the calculation of the timing of eclipses a lot more complex (for primitive people) as the planet would be moving more slowly at aphelion and much faster at perihelion rather than be moving at a uniform rate as would be the case in a circular orbit.
One further case for a person not standing at the equator:
As an observer moved north or south of the equator the arc of the stars passing across the sky would occur at a decreasing elevation until at the poles all of the stars would appear to move around the horizon in permanent although probably fairly bright twilight.