He didn't need to tell Dumbledore his suspicions because Dumbledore was already well aware: it was Dumbledore who asked him to keep an eye on Quirrell. It's explained in the last book.
The important thing here is that Quirrell behaved very differently before Harry's first year. When Quirrell takes a trip to Albania, the last rumored location of Voldemort, and comes back a stuttering mess, Dumbledore probably instantly knew what was up. He just couldn't catch him in the act.
This is explained in a conversation between Severus Snape and Bellatrix in half blood prince. Snape suspected Quirrell and threatened him. Quirrell knew it was Severus who interfered with a counter-curse when he tried to throw Harry off the broomstick.
The truth came out when Professor Quirrell admitted to Harry that he'd jinxed the broom and claimed he'd have gotten away with it too if it hadn't been for those meddling counter-curses. In other words, Snape tried to save Harry's life, and the only thanks he got was being set on fire. Harsh.
He explains also that Quirrell could not touch Harry because Harry was protected by his mother's love.
Quirrell's body manifests burns and blisters during his fight with Harry due to the protective power Harry's mother left in his skin when she died for him.
The scar is also a symbol of Harry's emotional sensitivity, because it hurts him whenever hatred is directed at him, as when Snape first sees him at Hogwarts or when Quirrell tries to grab him.
Snape enters Umbridge's office, and Harry yells “He's got Padfoot at the place where it's hidden!” in hopes that Snape will contact someone in the Order.
Quirrell knew it was Severus who interfered with a counter-curse when he tried to throw Harry off the broomstick. And Voldemort was there under his turban so he probably witnessed it too.
He was afraid that if he revealed himself to Snape, Snape might turn him over to Dumbledore and/or the Ministry and Voldy wouldn't be able to defend himself given his weakened state.
Snape's fierce devotion to and love of his childhood friend Lily, Harry's mother, is the foundation of that loyalty. After Harry Potter and the Half-Blood Prince, Snape's loyalty was a matter of intense debate among the fans.
Quirrell didn't use the Killing Curse on Harry, because Voldemort said “use the boy” to get the Sorcerer's stone from the Mirror of Erised. If Quirrell had killed Harry right away, he would have lost the chance to get the stone.
Hagrid later becomes friends with Ron and Hermione as well. Later in the book, a hooded person (Quirinus Quirrell in disguise) gives him a dragon egg to elicit details about Fluffy.
Voldemort's past, as revealed by the books, is shrouded in mystery and guesswork. In fact, at the beginning of the series, barely anyone but Dumbledore and Hagrid knew that Voldemort was the charming, successful Tom Marvolo Riddle.
Dumbledore knew that destroying Voldemort's horcruxes was the only way to truly defeat him. He believed that Harry was the one person who was smart, courageous, and perhaps lucky enough to find and eliminate all the horcruxes.
In Goblet of fire, Severus Snape apparated to the graveyard, 2 hrs later after Voldemort's failure to kill Harry. He convinced the dark lord that he had no idea about Voldemort being a host in Quirrell's body, and he was trying to stop Quirrell from taking the stone for himself.
In order to conjure the avada kedavra curse, you have to want to kill your victim. We all know that Voldemort could easily kill a child without an ounce of remorse... but not Snape. Snape didn't want to kill Dumbledore, and this was why the spell was blue instead of the usual green.
Voldemort intentionally made six Horcruxes, but when he used Avada Kedavra on Harry, he unintentionally created a seventh Horcrux. Instead of dying, Lily's love for Harry created a counter 'curse' known as Sacrificial Protection and saved Harry.
Dumbledore trusts Snape because he witnesses Snape's emotional devastation upon learning of Lily's death. Dumbledore himself knows that sort of devastation. It's akin to what he himself experiences when he contributes to his sister Ariana's death.
Image via Warner Bros. Similarly to Harry, Snape's patronus represents a person. In the memories he gives Harry before his death, Snape shows Dumbledore his patronus, using the form to prove his love of Lily Potter, Harry's mother.
He said, "According to Victorian flower language, asphodel is a type of lily meaning 'my regrets follow you to the grave' and wormwood means 'absence' and also typically symbolizes bitter sorrow." If you combined that, Snape's words mean "I bitterly regret Lily's death."
Lily was a muggleborn - to her, she was a mudblood. James married Lily and they had Harry, who is a half-blood. A pureblood is someone who has two pureblood parents. A muggleborn is someone with muggle parents, and a half-blood is someone with one pureblood parent and one muggleborn parent.
The reason why Quirrel couldn't touch Harry at the end of the first book is because Harry's mother Lily left behind a powerful magical enchantment when she died that protects him from Voldemort, who was possessing Quirrel at the time.
'The Prince's Tale'
Professor Snape is dead on Voldemort's orders, and Harry saw it all. In his dying moments, he told Harry to take his memories and look at him one last time. Voldemort's voice then sounded through the corridors, challenging Harry to meet him in the Forbidden Forest in an hour. This is the end.
According to the author, Severus Snape's birthday is 9 January 1960 and he died on 2 May 1998 at the age of 38.