"We might spend the evening together."
Translation:Nous pourrions passer la soirée ensemble.
la nuit is "the night", generally the period in which you sleep. le soir and la soirée both refer to the evening (though soir can sometimes translate to night as in hier soir → "last night").
soir indicates a simple amount of time or division of time, and is the period after 6pm to about midnight. You would say "Bonsoir !" to greet someone afer 6pm.
soirée indicates a duration of time, usually stressing the actual length of time. To wish someone a "good evening" when leaving them you would say "Bonne soirée" which means "Have a good evening" (which may not actually end at midnight).
If a man said "Nous pourrions passer la nuit ensemble" to a woman he didn't know very well, he might expect a slapped cheek in return.
I believe that the use of 'might' here is a bit off as a translation for 'nous pourrions' ... I would rather say "we could... "
'We might spend the evening together' sounds like something you'd say in response to someone's question. (not for every case though, of course)
+Are you coming back home later today? -Not sure, I am with XXX, we might spend the evening together.
I think the nuance there would be closer to: "peut-être passerons-nous la soirée ensemble"
'We could spend the evening together' sounds more like you are trying to make a suggestion to the person you are with.