For a start both (or all in multiplayer) players have a Frodo and a Ring, and each players Free People's cards have no effect on the other players'.
Each player takes turns to be the Free People's player, and the other player(s) is the Shadow player. On each turn the Free People's player gets a chance to play more Companions (max 9, incl Frodo), power-ups (Possessions, Conditions, Artifacts) or Allies. The costs for each of these things go into a pool. Then they move on to the next site, and put some more points into the pool for the danger of the site they've moved to, and the number of Companions they have in play.
Then the Shadow player(s) get to use the points in the pool to play minions, which the Free People's characters have to fight off. If the FP player is feeling confident then they can move on a second time, at which point the Shadow player gets to play more minions (and keep the surviving ones). If they don't move on, or if they've moved their move limit then their turn ends and all minions in play are discarded.
The FP player wins if Frodo survives the skirmish at site 9, the Shadow player wins if he is killed, or corrupted by the Ring.