I'm writing a fantasy story set in a p-type circumbinary system and I'd like to know how this would affect day and night, when the suns and moon would be visible, and the climate and seasons on the planet.( Just a general answer please. Not too heavy on the technical aspects)
2 Answers
Two sunrises and two sunsets. Seasons are not specifically affected.
For a p-type system, a planet is orbiting a close binary pair just like it is a single star. According to Wikipedia, stable planetary orbit should lie at least 2-4 times farther away from the binary stars separation distance. This gives the maximum visible separation between them similar to the Sun-Mercury pair (Mercury has a semi-major axis of 0.387 AU). At times, the suns would be rising and settling as much as two hours apart, and at other times they would do it simultaneously. If the stars and the planets lie in the same plane (which would be likely), the suns would be seen "eclipsing" each other several times a year.
As for the seasons, two suns in p-system shouldn't make any specific impact. On Earth, seasons are guided by its axial tilt and, to much lesser extent, the distance to Sun. If this planet is set up similarly, axial tilt would have the same effect from 2 suns as it is from 1.
P.S. Sun-Mercury separation example of the farthest distance the stars can be seen from each other. The pair can actually be much closer and never seen more than a short span away from each other.

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If the stars are both similar to our Sun, then the habitable zone for an Earth-like planet would be (I think) roughly 40% farther away—i.e., √2 times farther—and their angular separation would be correspondingly smaller. (OTOH, if they're smaller stars, then the habitable zone might lie closer, perhaps even within the zone that's orbitally unstable.) – Michael Seifert Jul 23 '20 at 17:50
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@Michael Seifert yes, the planet should be farther from the suns - but then the suns can be farther away from each other. – Alexander Jul 23 '20 at 18:02
orbital stability is guaranteed only if the planet's distance from the stars is significantly greater than star-to-star distance.
So the pair of stars would approximate the course across the sky that our sun takes as viewed from Earth. Assuming the same axial tilt as Earth.
The suns would rise and travel across the sky at the roughly the same relative positions during the day assuming the day is short relative to both the year and the period of rotation of the suns around each other.
Over the course of many days the relative positions of the stars would change. Due to the stability consideration mentioned above they would remain in fairly close proximity in the sky (assuming a stable planetary system), but the distance between them would vary sometimes getting closer and sometimes further apart.
Assuming that the planet orbits in the same plane that the stars rotate (likely) there would be regular eclipses of one star by the other in alternating fashion.
Day length would be slightly longer on average due to the ability of the combined suns to shine over a little more than just one hemisphere. The day length would also vary a little depending on the phase of the suns (during an eclipse only one hemisphere can be illuminated as with our sun).
Seasons would depend on axial tilt, the size and relative positions of the suns and the planets orbital distance, but assuming Earth like circumstances (except with two suns) the seasons should be roughly the same as on Earth provided the Stars were of the correct luminosity.

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If you have more than one moon make sure the orbits are stable. – M. A. Golding Jul 23 '20 at 18:37