A Link to the Past - Sword Upgrades Guide

Getting the Golden Sword is not essential, but it is very helpful. The Golden Sword is a powerful weapon that deals four times the amount of damage as the White Sword does!
The White Sword is the default sword you start off with, whilst the Master Sword is obtained by gathering the three pendants in the normal world of Hyrule. Either way, both the White and Master Swords are obtained by default, thus the task at hand is to get the Tempered Sword and finally the Golden Sword.

Tempered Sword: Firstly, note that it is not essential for you to obtain the Tempered Sword in order to get the Golden Sword. However, getting the Tempered Sword is pretty easy, so it's worth spending a few minutes getting it. If you go into the smithy in Kakariko Village, a Dwarf smith will be inside. Talk to him and he'll tell you about his missing partner. In order to locate this missing Dwarf, head to the dark world version of Kakariko Village. Go through the southern exit of the village, and immediately head west. You should see a frog hopping up and down, surrounded by heavy rocks. Free him (you will need the titan's mitts), talk to him and he will start to follow you. The frog is in fact the missing Dwarf, transformed by the dark world. Head back into the village, then immediately east, where the abandoned house is. Use the magic mirror outside the house, and the Dwarf will return to normal as you escape the dark world. Go inside and the dwarves will be re-united, and as a reward they will offer to temper your sword. Go outside and after a few minutes they will be done with your new sword!

Golden Sword: The Golden Sword is a reward obtained from a fairy's fountain, located inside the Dark Pyramid. If you are willing to toss your Master Sword or Tempered Sword into the water, a fairy will appear and reward Link with the Golden Sword. However, the entrance to this fountain is sealed and the only way to get inside is to blow up the door with a Super Bomb. To get the Super Bomb, you must first obtain at least six of the crystals in the dark world. After that, head to the dark world bomb shop (it's in the same location as Link's house is in the light world). The 'Super Bomb' will now be up for sale, buy one. You can only have one Super Bomb at a time, and you must drag it behind you all the way to the Dark Pyramid to blow up the entrance to the fountain, so be careful not to set it off! The entrance is to the left of the pyramid, it's a large piece of stone with a crack in the middle. Once inside, throw your sword in the fountain and get your reward.
Note: You can also choose to throw your arrows in afterwards, which you cause you to be rewarded with the Silver Arrows. You need to obtain the Silver Arrows, as they are required in order to kill Ganon.