At the end of The Return of the King, it is seen that Gandalf takes Frodo away with him on some journey. Sam and others feel sad when they come to know about this.What is that journey? Why does Frodo have to leave Shire?
 Gandalf and evles leave for the world of Elves all the way west. The scar that the king of darkness gave Frodo (do you remember? Frodo was stabbed in The Fellowship of the Ring) cannot be cured in Middle Earth. That's why they take Frodo with them. 
Was Frodo an elf? If he was and all the elves left for the west, then why didn't Sam and others accompany him?
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Frodo isn't an elf, he is a hobbit.
Okay. By the way, why are the elves leaving with Gandalf?
I explained why they take Frodo with them. I guess you understand why Sam and other hobits stay. The reason why Gandalf leaves Middle Earth is that the time of Human has come. Other species such as Elf, Dwarf, etc are to disappear. And the Wizard's roll in Middles Earth is over. 
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Thank you, Inthemood.