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I never understood why Sauron was not looking for the Ring actively in the Lord of the Rings films. The Nazgul know that Frodo has the Ring and they should be knowing where Frodo is heading.

Saruman has sent his orcs to find the Ring, but why was Sauron not that interested?

The Ring was not hidden. It was in the open, as Frodo had used the Ring multiple times. The fellowship was broken quickly and Frodo did not have any special power to hide the Ring.

Is there any explanation of this in the books? It seems to me that the LotR movie series has omitted some important details about the relationship between the Ring and Sauron.

You'll find a lot of explanations on our sister site Science Fiction & Fantasy, e.g. Why didn't Sauron guard Mount Doom?. Those focus on the books, and most texts there didn't make the movies, but it can be explained from the movies alone, I think.

Basically, Sauron thinks everybody has the same intentions as he does, so the one wielding the Ring will use it to conquer Middle Earth. What feels more threatening, one or two little hobbits, or a combined army of Gondor and Rohan who just defeated Sauron's army at Minas Tirith? Plus, those hobbits could've joined the march to the Black Gate (they were last seen on the border between Gondor and Mordor, which is close enough to join). From Sauron's viewpoint that would make a lot of sense.

Plus, as @JoeL. puts it:

This was a key point of Gandalf and Aragorn's strategy, and the whole reason they led the army of Minas Tirith to the Black Gate of Mordor. The hope was to draw not only Sauron's attention, but his armies as well, leaving Mordor itself unguarded.

That's reflected in this scene after the battle for Minas Tirith:

Aragorn thinks Sauron will take the bait since he revealed himself as the heir to the throne of Gondor via the Palantir they snatched out of Orthanc.

Sauron had no idea what Frodo was going to do with the Ring, or what the Ring was going to do with Frodo. The fact that it was travelling to its doom did not occur to him.

Sauron assumed that the Ring would be used as a weapon of power, and was doing what he needed to do in order to counter that power. The fact that someone would seek to destroy it was inconceivable to him.

This is a key plot point: Sauron never even thinks of the possibility that the Ringbearer will try to destroy the Ring. He cannot imagine somebody voluntarily giving up that much power. From the book:

Gandalf: For he is very wise, and weighs all things to a nicety in the
scales of his malice. But the only measure that he knows is desire,
desire for power, and so he judges all hearts. Into his heart thought
will not enter than any will refuse it, that having the Ring we may
seek to destroy it. If we seek this, we shall put him out of his
reckoning.

After the Fellowship split, Gandalf and Aragorn work hard to maintain this deception. They want Sauron to believe that Aragorn has the Ring.

When Pippin uses the Palantir at Isengard, Sauron sees a hobbit for a few seconds. He thinks that Saruman has captured the Ringbearer.

After Isengard is destroyed, Aragorn uses the Palantir to taunt Sauron. From that moment, Sauron believes that Aragorn has the Ring.

Aragorn then does exactly what he knows Sauron expects him to do if he had the Ring: raise an army and march to the Black Gate to make a frontal assault on Mordor. This was a terrible risk, taken to keep Sauron's attention and forces focused on Aragorn.

And note one more point: Sauron was right. Nobody could voluntarily destroy the Ring. When the time came to throw the Ring into the fire, Frodo could not do it. If Sméagol had not been there, Frodo would indeed have succumbed to the Ring's temptation and tried to wield its power.

In a letter to a reader, Tolkien described what would have happened next. Sauron would have immediately sent the eight surviving Nazgul.

I think they would have shown 'servility'. They would have greeted
Frodo as 'Lord'. With fair speeches they would have induced him to
leave the Sammath Naur [Mount Doom] - for instance 'to look upon his
new kingdom, and behold afar with his new sight the abode of power
that he must now claim and turn to his own purposes'. Once outside the
chamber while he was gazing some of them would have destroyed the
entrance. [...] In any case a confrontation of Frodo and Sauron would
soon have taken place, if the Ring was intact. Its result was
inevitable. Frodo would have been utterly overthrown: crushed to dust,
or preserved in torment as a gibbering slave. Sauron would not have
feared the Ring! It was his own and under his will.
Tolkien Letters 246