If you're past the first mission and don't have a shotgun or assault rifle, then you chose the wrong class to carry those. Starter versions of each of the weapons available to your class are automatically given to you at the beginning of your first proper post-tutorial mission (i.e. And yes: it did tell you this when it asked you to pick a class during character creation, as well as which weapons were available to what classes. You can only use certain weapon types according to the character class you chose for Shepherd at the beginning of the game. New weapons are found on missions as scripted loot items.
