First off, you can't just use any plain old rice for sushi. Regardless of what ingredients you added to it, regular long-grain rice would just fall apart - not to mention why the heck go to all the trouble to create something like the artwork sushi is supposed to be using the wrong type of one of the most important ingredients?? It must be short-grain rice - preferably a Japanese variety & these days nearly all supermarkets will carry one or two of them in their Asian section. In fact, even places like CostCo carry it - albeit in larger sizes. But kept in an airtight container in a cool, dry place, it keeps very well, & can also be used to accompany other Asian dishes.
Second - you only need a bamboo mat & nori seaweed if you're planning on making Makizushi - which are the nori "rolled" sushi (although even there, creative sushi chefs will sometimes do them cubed as well). And if you do the large cone-shaped handrolls, you only need the nori - no bamboo mat. Only soak nori if you want to end up with a disintegrating slimy mess. It should be lightly toasted which should make it quite pliable. If too old to begin with or overtoasted, it will crumble. However, don't throw it away if this happens. Save it in an airtight container & use it to garnish other Japanese dishes - especially rice & soups.
For Nigiri sushi (the rice ovals with toppings), all you need is the properly prepared & seasoned rice, your toppings, & *your hands*. With this type, a small strip of nori is only used as a decoration once in awhile.
I do a lot of this cooking at home, & while I do have a cute little bamboo rice paddle for cooling the rice (an important step), as well as several bamboo mats, neither is necessary for turning out basic sushi. If you have any specific questions, I have a Japanese friend whose family owns & runs a sushi restaurant & catering company. I doubt there's any question she can't answer.
There also are quite a few websites devoted to making sushi, as well as a number of online companies that sell frozen sushi-quality seafood of all types. It's EXTREMELY important that any raw fish you use has been frozen at zero degrees for at least 48 hours to reduce the possibility of parasites.
Good luck, & have fun making your own sushi - I know I do!!!