I'd imagine the characters would try using the wand and, finding that the methods of activating it don't just come naturally to them, realize they have to use alternative methods (AKA Use Magic Device) in order to coax the wand into doing its thing.
The subject of whether or not someone is aware of what spells are on their spell list is something that's never been considered in a game I was in. That seems like the sort of knowledge that's too meta for characters to know about in-character, but what they can definitely know is whether or not a magic item works easily for them or if they have to try and force it. Otherwise, if you really wanted to have them actually roll something, a Knowledge skill of the appropriate type (Arcana for arcane, Religion for divine, Nature for druidic-divine) seems like it's fitting. I wouldn't make the check a difficult one, since this isn't the sort of thing I would want to pose a significant obstacle to the PCs.
Part of the issue here is that it's fuzzy whether or not classes are even a concept in-setting. It's easy enough to say wizard and have that mean something specific in setting, but it's harder with classes like ranger. Yet, at the same time, anyone who identifies a wand with Spellcraft knows what classes are able to use that wand without UMD, so maybe the distinction between classes in-setting is pretty clear? Really, I suppose it comes down to what the setting in question is (usually Golarion, when it comes to Pathfinder) but I think most people like to assume most game mechanic terms aren't used in-setting.
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I'd say you can recognize whether a spell is on your list intuitively once you know what the spell is (via whatever check). It feels right. Like something you could do if you really put your mind to it.
Otherwise a fair number of mechanics stop making sense. I mean, Rangers pick new spells every day. How do they do that if they don't know their options?
so that's how I'd handle it.
From your character's perspective, there is no "spell list". Certainly no "class list".
Whatever your class, some spells just make sense, and some don't. If you're an arcane caster, for example, then arcane spells make sense. You already know enough about magic to understand arcane spells, although those that are higher level than you can cast might still be too complex to work out the finer details - but you still understand the general concept. But for that same arcane caster, divine spells just don't make sense. He doesn't understand the mechanics.
Think of it like this:
If a math student looks at a new math problem he's never seen before, he is going to know what it is and how to solve that math problem. If it's some higher level math using a few symbols he doesn't recognize, he's still going to know that it's math and he will know that he can solve it once he learns the correct principles. But when he looks at a sheet of music, he might not even know what it is, but even if he does, he definitely knows that it's not math and that he can never make sense out of it no matter how good he becomes at math - he needs a whole different skill set to make sense out of that sheet music.
The Question here, would a 1st level Sorcerer with fire powers pick up a Wand of Beast Shape and just know that he would be able to use it, since it is on the Sorcerer spell list? That, to me, doesn't make sense
And yet the rules say yes, he can pick it up and just use it.
The same 1st level sorcerer could pick up a wand of Cat's Grace (a spell beyond his casting ability), and use it as well.
Heck, a 1st level Ranger can pick up the same wand of Cat's Grace and use it too, despite not being able to cast spells at all, having no caster level, and having the spells that he will eventually cast be divine rather than arcane.
Yes, classes are just packages of mechanics and don't necessarily exist in the game world. But the ability to use a wand is a mechanic, and comes with the class.
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Obviously the Character knows the spell, if he can already cast it. The druid knows the spell she is capable of casting, even if the character senses it differently. A Cure Light Wounds might be phrased "oh dear nature, give me the power to cloase their wounds" and the player - out-of-game- phrases it like "I prepare one Cure Light Wounds" or similar.
The Question here, would a 1st level Sorcerer with fire powers pick up a Wand of Beast Shape and just know that he would be able to use it, since it is on the Sorcerer spell list? That, to me, doesn't make sense
It does to me. He might not have thought so before he saw it performed, or touched the wand in question, but when he does, the magic just sings to him. He can feel the arcane resonance in his blood saying 'This could be yours. This could be a part of you'.
And similar things for other Classes, obviously.
Now, a Sorcerer with no Spellcraft can't figure that sort of thing out very well, because he'll never know what spell is being cast/used, but it's definitely how it works when he can figure out the spell.
In my games, a sorcerer could totally activate the wand without knowing what spell is stored inside, if the spell is on his spell list.
But honestly, who takes that gamble?