Luke uses the same lightsaber throughout the first two films in the Original Trilogy, the same one that Obi-Wan took from Anakin at the end of their duel in Revenge of the Sith and that Luke was then given in Ben's shack on Tatooine.
Eventually he loses it, along with his hand in a duel with Vader at the end of The Empire Strikes Back and constructs a second lightsaber sometime between the end of the second film and the start of the third.
The following images are from the Star Wars: Visual Dictionary. The first is labelled "Skywalker's Lightsaber", the latter "Luke's Second Lightsaber"
Obi-Wan wields the same lightsaber throughout A New Hope, one that he constructed between the end of Star Wars: Attack of the Clones and the start of the animated Star Wars: The Clone Wars TV show, evidently in honour of his becoming a Jedi Master.
As an aside, you may wish to note that Obi-Wan's lightsaber evidently survived the duel with Darth Vader. While there is no further Disney-canon information about it, there's a rich seam of "Legends" stories that detail its fate, passing through several hands before finally ending up back in Luke's possession as part of his New Jedi Order.