(I've tried to research this online but I'm a linguist, not a physicist, and most places that discuss this quickly get too technical for me. Simple answers will get tons of gratitude!)
I imagine two suns locked in a fairly tight orbit (for simplicity's sake let's say Sun 1 is the same size as our sun, while Sun 2 is twice the size of ours) with a planet that moves around both in a noticeable ellipse. In the absence of any tilt on the planet(?) my thoughts were that seasons would be determined by the distance of the planet from the suns, and I plotted out the course of one 'year' as follows.
Summer 1: Planet is at 3 o'clock, at one of the two points where it is nearest to both suns in its orbit. Days are long and the entire planet is warm.
Autumn/Fall 1: Planet is at 4-5 o'clock, moving further away from both suns (however Sun 1 is further away than Sun 2) Days steadily grow cooler as the season progresses.
Winter 1: Planet is at 6 o'clock, at its furthest point from Sun 1 and far from Sun 2 (in the sky Sun 2 eclipses Sun 1 at midwinter) The entire planet experiences mild winter.
Spring 1: Planet is at 7-8 o'clock, days lengthen and temperatures rise.
Summer 2: Planet is at 9 o'clock, at its other closest point to the two suns. The season is a mirror of Summer 1.
Autumn/Fall 2: Planet is at 10-11 o'clock, with Sun 2 further away than Sun 1. Days grow noticeably colder noticeably faster than in Autumn/Fall 1.
Winter 2: Planet is at 12 o'clock. This is the coldest winter, and the one where snow is likeliest to fall. Days will be dimmer and shorter than at any other time in the year.
Spring 2: Planet is at 1-2 o'clock, with days growing warmer again. A milder season than Spring 1.
...I'm sure I'm doing something wrong here, but I don't have the knowledge background necessary to figure out exactly what. If anyone could help me poke holes in this sequence and give any hints at what the actual order of seasons and variations might be, I would really appreciate it!!