Other mistake: When Ted and Robin borrow Marshall's Fiero and are sitting in the car talking as Ted drives, you can hear the beat of a song playing in the car that is not the Proclaimers' I'm Gonna Be (500 Miles), which in many future episodes is referenced as the only song that can be played in that car due to the tape being stuck in the tape player. (00:15:30 - 00:16:10)
Suggested correction: Maybe it's on the radio.
First Time In New York - S2-E12
Continuity mistake: At the airport, Lily and Robin are talking off to the side away from others. During the scene, the cameras shoot from in front of the two actors and from behind Lily. From the frontal view, Lily's two headbands are white and gold-jeweled. The white headband is closest to her forehead. When the camera shot switches from front to behind, the headbands are switched on her head. The gold one is now closest to her forehead and the white one is behind.
Suggested correction: If you look closely, you can see that the headband is actually symmetrical, so the black strap is on top and in the middle of a larger white headband and never changes. (camera angle makes it often seem both from the front as from the back as if there is no white part behind the black).