The pair have penned two-year deals with the club.
Bennett, 28, burst onto the scene as a 15-year-old at Derby and became the Rams youngest ever-player in 2011.
On joining, Bennett, who also currently holds the record for Derby’s youngest-ever goalscorer and went on to make 89 appearances for the Championship club, said: “I have had positive talks with the manager and liked his ambition and his vision for the coming seasons, so it was a no brainer for me.
“He wants to get the ball forward and wants strikers who buy into the work ethic and pressing, and that suits my game.
“My challenge is the same as it’s always been, do the best I can and play as much football as I can.
“When you get the team spirit and a group all pulling in the right direction you can create special memories.
Bradbury, 27, was offered the chance to remain at Cheltenham, but has elected to head to Yorkshire.
The left-footer made 88 appearances for the Robins.
On joining, he said: “I had three really good years at Cheltenham so it was a tough decision to leave, but after meeting with the manager, there was only one place I wanted to go.
“The club is moving forward and I’m excited to be on board, I’m 27 now and coming into my prime, so I want to show the fans what I’m all about.
“I’m fully committed, I’ll leave everything on that pitch, really aggressive in the air and just try to do everything to keep the ball out the net really!”
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