All you need is a Kyber crystal and a hilt. Once you have them both in your inventory, click the crystal and select "Install Crystal in Hilt". This will bring up a dialog that allows you to select an empty hilt. Select it and voila! You'll assamble your first lightsaber! :)
You can find crystals in scrap piles, and you can buy hilts and crystals from the lightsaber vendor in Black Spire.