What nights are the visible planets actually visible?
They're visible almost every day of the year.
While you'll often hear that one planet or another is visible for some limited period of time, it's generally just an "extra good" time to view them.
You'll often hear that a particular constellation or celestial object is a "Spring Constellation" or "Summer Target".
That seasonal designation just means that it's visible in the summer sky after sunset - the time when most casual observers are likely under the sky.
We should all understand that if the sun were not in the sky, we'd see 360 degrees of stars and planets. It's up there all the time.
The sun washes out the whole sky half the day, right? So we should only see have the sky on any give 24-hour day, right?
Actually, consider what happens when the sun goes down and it gets (mostly dark):
If you look up, you see half the sky. Now stand there for the next eight hours, what are you going to see? You are going to see a whole bunch of new sky rise. 1/2 sky plus 1/3 sky (8 hrs / 24 hrs = 1/3 rotation) = 5/6 of the sky. Depending on your lattitude and the season, you'll have a little more or less time before sunset: you can see most of the sky on any given night.
So to see any particular object, you just need to know what time it's in the sky and decide whether you're up for it.