Why did Gandalf take Pippin when saving Faramir and company?

You've hit the nail right on the head. According to Peter Jackson (in the film's commentary track), Pippin and Gandalf were originally going to have been just arriving at the city when they would have espied the riders returning. They would then have wheeled around and rode out to save them. Pippin, naturally would not have had time to dismount which is why he was on the horse with Gandalf.

Peter Jackson: So, this was a sequence where we had a little bit of trouble because we've put it in a completely to how it was shot, if you remember how Gandalf and Pippin arrive at Minas Tirith for the very first time and they get to the brow of the hill and stop and they look at the city.

What was going to happen was that they were going to see this retreat and they were going to intervene before they'd even gone into the city. We've now put the sequence in much later and had Gandalf leaving from the gates of the City itself.

Since those scenes had already been plotted, when the decision was made to have Gandalf arrive at the city and chat for a bit before riding out, it required him to collect Pippin again before he left, despite that not making a lot of sense in-universe.

In-Universe, there may have been any number of reasons why Gandalf took Pippin along with him to help the Gondorians.

Pippin had scarcely left Gandalf's side since they had arrived at Minas Tirith. Pippin had never been to a city of such size, the largest place he had been to was Edoras, which dwarfed in comparison. Maybe he didn't want to be left alone in such a big confusing place that he had only arrived at days earlier.

Likewise, perhaps Gandalf didn't want to leave him in such a place. Pippin was notoriously inquisitive and constantly getting himself into trouble (such as in the Mines of Moria, causing the Fellowship to get attacked when he knocked the skeleton into the well, or taking the Palantir and accidentally speaking to Sauron). It's possible that Gandalf didn't trust him to keep himself out of trouble whilst he was gone, thus thought keeping him close by would have been safer for him.

It's possible that Gandalf and Pippin were already on the way somewhere on the back of Shadowfax when they became aware of the soldiers fleeing Osgiliath, and he didn't want to waste anymore time getting Pippin off of the horse before going to the aid of Faramir & co.

Out of universe, it was so that Faramir could catch sight of Pippin for the first time in the presence of Gandalf, causing him to see the recognition in his eyes and coming to the conclusion that Faramir and Frodo had crossed paths at some point.

I can't remember how this interaction is approached in the books, but with the time constraints of the movies, I'm assuming that Gandalf learning the fate of Frodo needed to happen rather quickly, as it is in the following scene where Denethor is chastising Faramir for letting Frodo and the Ring go whilst Gandalf is present.

This scene would have made less sense if the audience had not seen Gandalf become aware of Frodo's fate beforehand, and was important to establish the relationship between Faramir and Denethor, who would realistically meet at the earliest opportunity, meaning that there would not have been the chance for another scene where Gandalf learns that Faramir has met Frodo before.

A plausible reason we could invent, after the fact, is that Gandalf used Pippin as a decoy ring-bearer, similar to how in the book Aragorn revealed himself to Sauron, unfurled the king's flag in battle, etc. Since Pippin had looked in the Stone of Orthanc, we hear Merry say "The enemy thinks YOU have the ring!" So we might pretend that in this scene, Gandalf and Pippin are furthering the deception in a continuation of the basic strategy to clear the way for Frodo and Sam, by keeping Sauron's attention focused elsewhere. The Great Eye sees all, so presumably Sauron was watching Pippin during the rescue,

In my opinion the fact that Pippin had looked into the Palantir was a huge factor in Gandalfs decision.

Gandalf needs to protect Pippin for a number of reasons:

1. Sauron is now 'aware' of him and, desparate to get the ring may send the Nazgul to get him.
2. As above, Gandalf also knows that Frodo and Sam are trying to sneak into Mordor and, Pippin revealing himself to Sauron (as a hobbit) will draw Saurons attention away from the borders of Mordor and deeper into Middle Earth.

So, Gandalf needs to keep Pippin close for protection. Gandalf will also feel responsible for the hobbits in a kind of 'proxy' way. Frodo and Sam were the only hobbits that Gandalf had 'officially sanctioned' into coming but, they are way beyond his influence. So, he's making sure that the hobbits he can protect are protected.