I totally agree with Dellwovideo, with one exception (and I say this with a lot of respect for his answer). I would be careful in slowing down the 60p video in a 24p timeline. Depending on the way your editing system interprets the native playback framerate of your 60p video file, it could automatically treat or convert your video as 24p when you play it back in a 24p timeline. This will result in a video that plays back at normal speed when you place it in your timeline and most importantly, strobey video when you slow down your 60p footage. Instead, I would make sure that your editing software is interpreting the playback framerate of your 60p video as 24 fps. This way, when you drop your video into your timeline or sequence, it will already be slowed down by 1.5x.
In order to get it to play at normal speed, you'll have to increase the speed to 250% (someone correct me if I'm wrong here, my math isn't always accurate). In order to get the video to play in slow motion, you can simply decrease the speed of the video and it should play back with a full frame for every frame per second in your video making it look great while getting rid of the strobey look.
If you're editing in After Effects, you can make sure that your 60p footage is interpreted for playback at 24 fps by right-clicking the video file in your Project window and choosing Interpret Footage and Main from the drop-down menu. Then under the Frame Rate heading, Choose the radio button that says "Conform to frame rate:" and type in 23.976 into the blank field next to it. Hit OK and you're good to go!
Unfortunately, I'm not sure how to do the same thing in Sony Vegas. Maybe someone else can chime in to help!