Learn English – If : second conditional or third conditional

A conditional sentence has two parts: the "if-part", or protasis, and "then-part", or apodosis.

In the apodosis, we use would to create that "conditional" feeling. So the sentence

If we had had lots of money, we should have travelled round the world.

would be illogical: the protasis clearly says that we did not have money during some period in the past, but the second half of the sentence is not an apodosis, because it uses the wrong modal verb. The second half looks like a normal sentence on its own:

We should have traveled around the world! (instead of doing some other things in the past, we should have traveled around the world)

This clause implies that we had the ability to travel around the world. It contradicts the protasis.

The same applies to your conditional 2 example:

If I worked harder, I should pass the exam. [improbable situation in the present or future],[moral obligation]

The first part invites some apodosis with would. The second part just plainly states your obligation to pass the exam. But according to Wikipedia,

Occasionally, with a first person subject, the auxiliary would is replaced by should (similarly to the way will is replaced by shall). (Wikipedia says this about the apodosis of both the second and the third conditional)

So maybe we can use should there after all, since I is a first-person subject. Let a native speaker decide.

The use of should is usually "deontic" (what should be: used to express norms, expectations, speaker's desire) while the use of would is "epistemic" (what may be).

The modal verb should could be sometimes used in the epistemic mood too, so we can come up with sentences such as

If I worked harder, that should be surprising!

But the effect would be comical and ironical, not the usual effect with the conditional sentences. The two halves of the sentence would still look somewhat disjointed.

Instead of future real / unreal conditions let's put it this way

Future likely / unlikely conditions.

When the condition is likely you use the verb in the infinitive form

For example

1. If you try to take pictures of restricted exhibitions, a member of the staff will ask you to put your camera away. (Here you think that the person whom you are telling this is likely to take pictures in the exhibition)

Now suppose you are addressing the same sentence to a group of people (where they are unlikely to take pictures inside the exhibition) then you use verb in the past tense i.e.

2. If you tried to take pictures of restricted exhibitions, a member of the staff would ask you to put your camera away. (It means I don't think you will try to take pictures)

Let's see few more examples

1. If he asks me, I will tell him. (I think he will ask me)
2. If he asked me, I would tell him. (I don't think he will ask me)

So using the verb in its infinitive form or in the past tense causes the meaning of the sentence to differ a bit.

Let's see your last sentence and the thinking that goes inside when you use 'will' or 'would' in the condition.

As we know 'would' is the past tense of 'will'. So the verb used in the past tense denotes less possibility of something happening. So when you say

1. It would be great if you could learn something every time you went to a museum. (It means you don't think the person will try to learn everytime he visits the museum)

But when you say

2. It will be great if you can learn something every time you go to a museum. (It means you think that he might try to learn everytime he he visits the museum.)

Let's take TRomano's example

1. It will be great if you can stop by and say hello when you arrive on campus. (It means i think you will come)
2. It'd be great if you can stop by and say hello when you arrive on campus. (I don't think you'd come)