Althoguh nintendo has made an official zelda time line, mario timeline is still a bit unconfirmed. If going to explain my guesses about this timeline. Note: I am not going to put DK, Wario or Yoshi games on this, just Mario:
-Mario Bros. (MB): This game and the cartoon opening happen at the same time, telling us the story of how the Mario Bros. got into the mushroom kingdom. They were sucked by a drain in a bath and inside found some monsters: shellcreepers, sidesteppers and fighting flies, but they were the best plumbers in the world. At the end of the pipe they found...
-Super Mario Bros. (SMB): A mysteryous world with mushroom habitants. One of them called Toad told them that many mushroomers got turned into blocks by the evil king koopa Bowser. Princess Peach tried to save them but Bowser kidnapped her. Mario and Luigi decided to rescue the mushroomers and as a way to say thanks they gave him mushrooms, as food provision and fire flowers, to make them able to shoot fire. Eight of them werent turned onto blocks, these were the mushroom retainers which you save in each world. When they arived to Dark Land, they managed to defeat Bowser and save Princess Peach, which gave Mario a Kiss as a thank you.
-Super Mario Bros: The lost levels (SMBTLL): When Bowser captured Princess Peach previously, he discovered that he had feelings for her, so decided to do that one more time, but his time sending more of his elite troops, thats why we see more hammer brothers and red piranha plants in this game. He also captured even more toads as we can see in Super Mario Allstars. Mario and Luigi had to come to a similar adventure than the previous and defeated Bowser one more time.
-Super Mario Bros 2 (SMB2): After a long adventure, Princess Peach constructed a house for Mario and Luigi in thanks. Mario went to sleep, but he had a strange dream. He was climbing stairs and found at the end a strange door. He heard some voices that asked him for help, and being the hero that he is decided to do it, but he awakened. Later he, his brother, the princess and Toad found a cave, which there was the land of Mario dreams, Subcon. These 4 decided to free the Subcons, the ones who were calling for help from the one who was imprisioning them, Wart Koopa, Bowser´s brother. After a long adventure they defeated him and saved the Subcons. Just to clarify the things the whole adventure wasnt Mario´s dream, there are two ways to arrive to Subcon, throught a Magical Book or throught dreams. The proof that it wasnt a dream is...
-BS Super Mario USA (SMUSA): After the previous adventure, Mario and his friends thought it was all a dream, but Wart stroke again placing all his minions into different places and stealing all the gold statues that the Subcon King made for Mario in thanks. Mario´s gang somehow was absorbed here again and defeated Wart once more. King Subcon thankful gave Mario a copy of the magical book called enclosed instruction book which said how to go inside warp zones and arrive to Subcon when he wanted, and many more things that could help on his adventure.
-Super Mario Bros. 3 (SMB3): Just when the mushroom kingdom thought it was at peace, Bowser stroke with a bigger menace: his children, the Koopalings. They stole the wands of the kings of different lands and turned them into animals. Mario and Luigi traveled to these lands, while Toad gave them different powerups inside chests. Mushroom kingdom resources were low at that time, thats why Toad just let Mario brothers take one chest. Once they rescued the king of Pipe land Bowser revealed his true plans, to kidnap Princess Peach... again. Mario and Luigi arrived to Dark Land one more time, defeated Bowser and rescued Princess Peach again. The proof this happened before Super Mario World is that Wendy mentioned it was her first time fighting Mario in the manual.
Note: Princess Peach isnt the mother of the koopalings, their mother is another female Koopa named Clawdia.
-Super Mario World (SMBW): Mario, Luigi and Peach went on vacations to Dinosaur land, while they werent paying attention, Princess Peach was thought to have been disapeared, but later found her inmidiatly. They noticed there were stone houses and then some koopas attacked them. Luigi fought face to face, with what he called a Fire Sumo (the true name was Sumo Bros, Mario knew that, but Luigi not). Luigi fell into the underground for hours, while Mario found an adult Yoshi (this was the Yoshi of Yoshi Island) who asked for help because his friends were imprisioned into eggs, he gave him a ride for the second plane of Yoshi Island. Mario defeated Iggy, Morton and Ludwig and found Luigi, who had a secret weapon, a baby Yoshi which would help them on later adventures. They continued and noticed he didnt like to enter castles or ghost houses. They later defeated Bowser and the rest of the koopalings and saved princess peach one more time.
-Super Mario 64 (SM64): After the adventure in dinosaur land, and the first time Bowser kidnapped the castle itself (gonna be explained later), Peach invited Mario, Luigi and Wario to a party, but somehow, they ended being kidnapped by Bowser minions. A friendly Lakitu, woke up Yoshi, who would rather sleep in the roof than be at the party, after telling what happened to his friends, Yoshi decided to adventure himself and rescue his friends. He found out that Mario was kidnapped by King Goomboss, which was angry for all the goombas being stomped all the time, and seemed to already have a grudge on him. Mario was freed from Goomboss power, and he and Yoshi went to fight King Boo, who had Luigi imprisioned. This was the first time King Boo meet the Mario brothers. Last but not least they knew they needed to team up with Wario if they wanted to defeat Bowser and recover all the stars from all the areas, so they defeated Chief Chilly, a giant Chill Bully with moustache. Once everyone was safe, they confrontated Bowser 3 times, managing to win in all of them, and managed to rescue Princess Peach, last but not least, they found all the stars, celebrating with a cake.
-Super Mario Sunshine (SMS): After that whole incident, Mario and Peach decided to take vacations to Delfino island, but Luigi didnt went with them because he was visiting Princess Daisy in Sarasland. Someone (besides those game theorists -_- ) was blaming Mario for doing bad actions, like ruining the island with graffiti. Princess Peach was surprised about that, since Mario had been with them the whole time. After they arrived, they realized that the one blaming Mario was none other than Bowser`s youngest child, Bowser Jr., who kidnapped princess Peach believing she was his mother because of Bowser`s lies. To find Bowser, Mario had to go on a similar quest than before and find the shine sprites, with the help of a new partner, the FLUDD. After finding Bowser, cleaning the island, defeating Shadow Mario/Bowser Jr. and recovering the shine sprites, Mario fought him and ended victorious once more. Bowser, after his failure, said sorry to his son, but then he revealed that he knew Peach wasnt his mother, he just wanted to fight Mario to make his father proud. Meanwhile, Mario tried to fix a hurt Fludd, luckily, E. Gadd, the Fludd`s creator, managed to help him.
-New Super Mario Bros (NSMB): After the events on Delfino Island, Mario and Peach returned to the Mushroom Kingdom, but unknown to them, Lakithunder, a dark Lakitu, started creating storms around the kingdom, what was worse, Bowser Jr. entered and kidnapped Princess Peach (TBC)
-New Super Mario Bros 2(NSMB2)
-New Super Mario Bros Wii (NSMBW)
-Super Mario Galaxy (SMG)
RPG games, SPP, Luigi Mansion and recent games coming soon