Starting Out
The number of armies you begin with depends on the number of players you have. For a three-player game, each person starts with 35 armies; four players have 30 armies each, five players have 25 and six players have 20. Deal the Risk cards to determine your starting territories, then place one army in each territory for which you hold a card. Roll to see who goes first, then distribute armies one at a time, in order, until all of your allotted armies are on the board.
Objective and Basic Gameplay
The goal of the game of Risk -- without missions -- is to control the world. You do this by owning every territory. On your turn, take the white attacking dice and declare a move, saying "Attacking this territory from here." Give the red defending dice to the player whose territory you are attacking. The attacker rolls an equal number of dice for the armies he wants to attack with, with a maximum of three armies/dice, and the defender rolls either one or two dice, depending on whether he wants to defend with one or two armies. Compare each side's highest dice; the loser loses an army. Compare the second highest dice; the loser loses an army. Ignore any extra attacking dice past the defender's last die. If the attacker has only one army left, the attack ends. If the defender loses all armies, the attacker can move armies into the new territory.
Missions
Your game should have come with a set of mission cards. Although you don't have to use these, they make for much shorter games. Keep your mission a secret and set out to do what the card directs. When you have finished your objective, on your turn, flip the card over and declare victory.