Having elected to field, once Palladino had bowled Steven Mullaney for 76, Notts wickets fell regularly.
Palladino also removed Che Pujara (29) and Samit Patel (28) before Rikki Wessels' 39 helped Division Two leaders Notts to a batting point.
The visitors then lost Luis Reese and Shiv Thakor, but got to stumps on 52-2.
Notts, unbeaten in their five County Championship games this season, endured a testing, slightly rain-affected day against their neighbours, whop began this match winless in ninth place.
Tom Taylor (2-58) and Conor McKerr (2-53) backed Palladino up well with the ball, helping Derbyshire enjoy the best of the opening day, despite Thakor's dismissal for 13 with just two balls remaining.
Derbyshire seamer Tony Palladino took 4-44 to help his side bowl out Nottinghamshire for 229 on the first day at Trent Bridge.