Is there offsides during direct and indirect kicks after a penalty? I thought there was, but was told differently today. Thanks for helping.

Assuming you mean foul or infraction by penalty, then the attacker can not be offside on a free kick, direct or indirect. An attacker can be in an offside position only if receiving the ball directly from a throw in, corner kick or goal kick. An attacker in an illegal position on a penalty kick or kickoff is guilty of encroachment, not offside.

Two things:
1. offside is singular, please do not put an "s" at the end.
2. yes, the attacking team can be penalized for this when taking a free kick, whether direct or indirect.

Just to clarify what Cecil said: If a player is in the offside position; he can legitimately receive the ball directly from a throwin, goal kick or corner kick. At no other time during the game - including other free kicks - can a player in the offside position legitimately receive the ball from a teammate.
A player can be in the offside position for the entire game, as long as he is not actively involved in the play.

Thank you!