You cannot play on another person's spread until you have laid down your own spread. All of the objectives in the game include assembling sets and runs. A set is three or more like-valued cards of any suit, such as three sevens. To buy a card, pick up the discard and one unknown card from the deck. You are not entitled to make a discard when making this play.
Play continues with the person after the one who made the discard.
Each discard is only available once and not whenever it happens to be the top discard. This means that after a discard is purchased, the person whose turn it is must draw a card from the deck. We think this makes the game more challenging and thus more enjoyable. You may wish to change this rule. They can replace any missing card in a set or run, provided there are not too many wild cards.