Related: Why did Bellatrix rely on torture and the word of a goblin to get information from Hermione, and not use the Imperius curse?

Very loosely related (although a lot of the reasons given there don't apply here): Why didn't Bellatrix use Legilimency on Hermione?

Why did Voldemort torture Ollivander for information on the Elder Wand, given the existence of clear alternatives? There's a fair amount of evidence that torture tends to produce unreliable intelligence.

Why didn't he simpy use Legilimency on him to read his mind? I'm not aware of any evidence that Ollivander knew Occlumancy. Wouldn't that have been faster?

Couldn't he also have used Veritaserum? Based on the fact that he invented a potion to resurrect himself, he appeared to have been a skilled potion maker, so couldn't he have made it himself? He presumably could've asked Snape to brew it for him, too. I do recall the book saying that Veritaserum was a legally controlled substance, but that doesn't seem to have been much of an obstacle to him in other ways. Given that he had apparently read quite a few books in the Restricted Section of the Hogwarts library, it would be hard to believe that he had never encountered any information on how to make it himself. Given his propensity to manipulate others, this seems like exactly the kind of thing that Voldemort would want to know about.

Both of those methods would likely have been faster and (arguably) more reliable than torturing Ollivander.

So, why did he torture Ollivander?

You raise two particular methods that Voldemort could have used but didn't. Why he specifically used torture in this instance is never directly addressed in the books, but we can attempt to use other information to address the question.

Veritaserum

I would argue that there are three factors that might make Voldemort hesitant to use Veritaserum to extract information from Ollivander:

• 
  

  It takes quite a long time to brew.

  
  
  

  In Order of the Phoenix when Umbridge asks Snape for more Veritaserum, the response is:

  
  
  

  
  

  “Certainly,” said Snape, his lip curling. “It takes a full
  moon cycle to mature, so I should have it ready for
  you in around a month.”

  
  

  
  
  

  Assuming Voldemort did not have a stash of Veritaserum on hand, he may not have wanted to wait a full month before acquiring the much-needed information from Ollivander.

  
  


• 
  

  It is not perfect.

  
  
  

  In Half-Blood Prince Harry asks Dumbledore why he doesn't just use Veritaserum to get Slughorn's memory, and Dumbledore responds that Slughorn might use an antidote:

  
  
  

  
  

  "But surely, sir," he said, keeping his voice as respectful as possible, "you don't need me — you could use Legilimency... or Veritaserum...."

  
  
  

  "Professor Slughorn is an extremely able wizard who will be expecting both," said Dumbledore. "He is much more accomplished at Occlumency than poor Morfin Gaunt, and I would be astonished if he has not carried an antidote to Veritaserum with him ever since I coerced him into giving me this travesty of a recollection.

  
  

  
  
  

  While there would not necessarily be a reason for Voldemort to suspect Ollivander of having an antidote, and presumably the antidote would wear off eventually, it is possible that Voldemort took this factor into account. Additionally, though there is no real evidence for this in the books, JK Rowling has claimed that Veritaserum is not infallible.

  
  
  

  If there was some way for Ollivander to potentially counteract the Veritaserum, not only would Voldemort not get the information he needed, he presumably wouldn't even know that Ollivander was not telling the truth and would thus be worse off than had he not used the Veritaserum. By torturing him instead, Voldemort can simply continue the torture until Ollivander gives him the information he wants. While there would still be no guarantee that the information was truthful, Voldemort could assess the information and possibly figure out whether it is true or not, and he could also test out the information and then renew the torture if the information turned out to be false.

  
  


• 
  

  Veritasereum is an external force.

  
  
  

  If we are to trust Dumbledore's assessment of Voldemort's character, Voldemort is obsessed with his own superiority and does not trust, nor wish to rely upon, anyone else. Having someone (e.g. Snape) brew him Veritaserum would be trusting and relying upon someone else. Even making it himself would be relying on the Veritaserum. Using a potion to extract information would be almost an implicit admission that he could not accomplish this goal with his own magic but had to resort to a potion that anyone could use. Torturing Ollivander, on the other hand, would be a mighty display of his own magic and would be overpowering the enemy by his own prodigious magical skill. This, then, may have been another factor in the decision to not use Veritaserum.

  
  

Legilimency

Using Legilimency would presumably not involve the first and third factors mentioned above regarding Veritaserum. Legilimency would be Voldemort's own magic rather than an external magical force, and it takes no time to prepare. However, the second factor mentioned above is even more prominent when it comes to Legilimency than by Veritaserum. Throughout the final three books we are told about Occlumency being an effective method of repelling Legilimency. While Ollivander is never specifically described as being an accomplished Occlumens, we also don't really find that Occlumency is a particularly advanced skill. Specifically, regular students were assumed to be capable of learning and mastering it. While Harry never did really master it Draco Malfoy apparently did, at least enough to block Snape's Legilimency in Half-Blood Prince:

"Who suspects me?" said Malfoy angrily. "For the last time, I didn't do it, okay? That Bell girl must've had an enemy no one knows about — don't look at me like that! I know what you're doing, I'm not stupid, but it won't work — I can stop you!"

There was a pause and then Snape said quietly, "Ah... Aunt Bellatrix has been teaching you Occlumency, I see. What thoughts are you trying to conceal from your master, Draco?"

Thus, there is no particular reason to think that Ollivander could not have employed Occlumency. Therefore, using Legilimency against him would not necessarily result in any useful information. Moreover, as mentioned above regarding Veritaserum, it is possible that Voldemort would not be able to tell that Ollivander was using Occlumency and therefore he wouldn't even know whether Ollivander was concealing additional information.

Conclusion

It seems possible that there may have been various factors causing Voldemort to specifically use torture against Ollivander. Some of these factors might include the desire to use his own magical power, the desire not to delay, the desire to be sure that his results were accurate, and the desire to be as foolproof as possible.

This all assumes that Voldemort in fact did not use Veritaserum and Legilimency on Ollivander. In truth, though, there is not really any evidence that he didn't. While Veritaserum is something that you might expect Ollivander to have mentioned if it had been used, Legilimency is not. In fact, Ollivander would not have necessarily even known if Voldemort was using Legilimency. All we know for certain is that Voldemort tortured Ollivander. However, it is certainly possible, in fact probable, that he also used Legilimency to help determine whether Ollivander's statements under torture were true.

Because he's a sadistic jerk. Why buddy up to someone you don't care about to get information that, unknown to to you -- because you don't care -- could likely be obtained with just a little time and effort, when self-gratifying fear and intimidation and a little adrenaline-inducing enjoyment of torture will do?

Remember, Voldemort isn't a psychopath; he is entirely controlled by his emotions, and we know he has derived sick pleasure from frightening others since childhood. He is also formidably intelligent, and knows how to develop and apply strategies to obtain what he wants, at least as often as he doesn't blow it by having a temper tantrum.

He is also a bit of an idealist. We have (at least here in the USA) a similar problem with the idea of torture. It is regularly presented as good and useful in films and books, deeply believed to be just and effective by about half the population -- all without regard to actual data and expertise by people whose careers are about obtaining information from enemies.

Voldemort enjoys terrorism and torture and is easily manipulated by his own pleasures in the moment. Tell me what I want or suffer, muah ha ha ha!

He tortured Ollivander partly out of rage.

The reason the Dark Lord needed Ollivander was because his wand didn’t work against Harry - something which he’d be very angry about. Once he had Ollivander, it’s likely that his choice of method was partly based on the anger he felt at the situation. He tends to lash out when he’s angry, whether or not it’s otherwise useful.

“The Elder Wand slashed through the air and green light erupted through the room, the kneeling goblin rolled over, dead, the watching wizards scattered before him, terrified: Bellatrix and Lucius Malfoy threw others behind them in their race for the door, and again and again his wand fell, and those who were left were slain, all of them, for bringing him this news, for hearing about the golden cup –”
_{^{- Harry Potter and the Deathly Hallows, Chapter 27 (The Final Hiding Place)}}

It’s likely that he chose to torture Ollivander as both a way to get information, and because he was sufficiently angry that he might overlook other options to choose torture.

“He wanted you to tell him how to overcome the
connection between our wands,’ said Harry.

Ollivander looked terrified. ‘He tortured me, you must understand that! The Cruciatus Curse, I – I had no choice but to tell him what I knew, what I guessed!”
_{^{- Harry Potter and the Deathly Hallows, Chapter 24 (The Wandmaker)}}

After that, the Dark Lord was torturing him because he’d suspected Ollivander was lying because using a different wand hadn’t worked.

“You told me the problem would be solved by using another’s wand!’

And into his mind burst the vision of an emaciated old man lying in rags upon a stone floor, screaming, a horrible, drawn-out scream, a scream of unendurable agony …

‘No! No! I beg you, I beg you …’

‘You lied to Lord Voldemort, Ollivander!’

‘I did not … I swear I did not …’

‘You sought to help Potter, to help him escape me!’

‘I swear I did not … I believed a different wand would work …’

‘Explain, then, what happened. Lucius’s wand is destroyed!’

‘I cannot understand … the connection … exists only … between your two wands …’

‘Lies!’

‘Please … I beg you …’

And Harry saw the white hand raise its wand and felt Voldemort’s surge of vicious anger, saw the frail old man on the floor writhe in agony –”
_{^{- Harry Potter and the Deathly Hallows, Chapter 5 (Fallen Warrior)}}

At that point, the Dark Lord was very angry, and likely tortured Ollivander out of rage as well as his need to find out the correct information about the wands from him. The other information he wanted from Ollivander, like about the Elder Wand, he wanted after that.

Because he did not know what to look for

Voldemort does not have any qualms about using legilimency. He uses it on Gregorovitch after Gregorovitch tells him that the Elder Wand was stolen from him. So it is easy for him to figure out what to look for. A memory of some young boy stealing the Elder wand, which must be one of Gregorovitch's most prominent memories. When Snape penetrates Harry's mind, he sees a lot of flashes of memories of embarrassing/sad/happy important events. So basically, looks like legilimency uses memories to detect contraditions and info. But it cannot "read" knowledge unless it is stored in a memory.

However in the case of Ollivander, the information he gets is about a "rumor". And rumors are not usually written down or easily visible or stored in some visual memory which could be easily accessible. So Voldemort used torture to "persuade" Ollivander to give whatever information he had on the Elder Wand.