As mentioned above, the earth's tilt remains fairly constant at 23.5 degree, no point in the 26,000 year cycle is much more favourable than any other.
The tilt of the earth produces seasons and the phenomena of equinox (length of day = length of night) and solstice (shortest / longest day).
Axial (equinoctal) precession will affect these as follows.
For the northern hemisphere, spring starts with the vernal equinox, in June, when the North Pole is towards the sun. But, because of the effect of precession, the point of the earth's orbit where this happens gradually changes. In 6,500 years this point will be about 90 degrees away from the current point and so occur three months earlier. Therefore (unless our calendar is amended) the Spring Equinox will occur in March. After another 6,500 years the Vernal Equinox will have slipped a further 90 dgrees or three months to December. And after another 6,500 years, to September. After the complete cycle of 26,000 years, the Northern Hemisphere Vernal Equinox will be back in June where it started.
Obviously this applies to all equinoxes, solstices and seasons for both hemispheres.
So eventually each month will be spring, then summer, then autumn, winter and spring again. But, ignoring other effects on weather, for any place on earth each season will be much as it is now.
The distance of the earth from the sun has very little effect on climate, seasonal changes result almost entirely from the different angles at which sunlight strikes the earth's surface.
(I know that some claim different ways of defining the start of seasons but pointing that out on this thread won't help anybody)