Firstly, you should rinse the sushi rice in a fine mesh strainer until the water runs clear. In a medium saucepan, combine the rice and water and bring to a boil over high heat. Thereafter, reduce the heat to low, cover, and simmer for 15-20 minutes, or until the rice is tender and the water has been absorbed.
While the rice is cooking, make the sushi vinegar by combining the rice vinegar, sugar, and salt in a small saucepan. Heat over medium heat until the sugar and salt have dissolved, then set aside to cool.
When the rice is done, transfer it to a large bowl and add the sushi vinegar. Use a wooden spoon or spatula to gently mix the rice and vinegar together, being careful not to mash the rice. Let the rice cool to room temperature.
Place a sheet of nori on top of the bamboo sushi mat, shiny side down. Wet your fingers with water and spread a thin layer of rice over the nori, leaving a 1-inch border at the top.
Arrange slices of salmon on top of the rice. Use the bamboo mat to roll the sushi tightly, using the border of rice at the top to seal the roll.
Repeat with the remaining nori, rice, and sushi fillings.
Use a sharp knife to slice the sushi rolls into bite-sized pieces. Serve with wasabi and soy sauce (you can also serve with other sauces which you desire, such as pickled ginger).