You question begs a question from me. Do you understand frame rates? Why do you want to do that and what do expect to happen? Last question, what kind of 24fps footage do you have?
Let's talk about kinds of 24 fps footage. First, there's real 24 fps footage, then there's 23.976 fps footage which was really 29.976 (ntsc) video which was recorded interlaced (59.97 fields per second) but has had 3:2 pulldown added which you typically get from a little older video camera with a 24fps option, then there is film originally shot at 24 fps but transferred to video using 3:2 pulldown.
Let's talk about 60 fps as a playback format. First, if you video is ever going to hit television of cable then there is no such thing. Everything in NTSC land is 29.97 fps then interpreted as 23.976 and it's all interlaced. What that means is that if you send progressive footage to a broadcaster you get interlaced footage back but it looks progressive because you have pairs of identical fields. If you live in PAL land you get other options. If you are talking playback on the internet via YouTube or vimeo, the standard is still 29.97, 25, 23.976 or 60P if the receiving client has 60p equipement.
If you truly have 24 fps footage with each frame a unique slice in time then when you "convert" that footage to 60 fps there are only two options. The first option is to just play back the footage at 60 fps which means 10 seconds on the clock will playback in 4 seconds so the motion will be sped up. 240 frames divided by 60 = 4.
The second option is that you keep the playback time the same for your 10 second clip by playing back some of the frames more than once. You can either do that by adding a pull down scheme or using time remapping or a plug-in like Twixtor to take the pixels in 2 frames and calculate their new position in the frame based on their projected movement. For example, let's say frame 1 has a rotating pointer that is vertical and pointed straight up and in frame 2 the pointer is horizontal and pointed to the right. If you were converting to 48fps instead of 60 then you would need a new frame between 1 and 2 with the pointer pointed up and to the right at 45º. This is very difficult for software to do with 100% accuracy, especially if there is a lot going on in the scene.
The only other scenario that I can imagine is that you have some true 60 fps footage that you want to use in the same project as your 24 fps footage. If that is the case, then the option I would pursue would be to create my project at 30 fps and use the 60 fps footage in that project and also import the 24 fps footage into the same project. If you have 60 fps footage that is really 60i or 60p then the frame rate is 59.97fps anyway and you could treat the footage as 29.97 with no degradation in quality or frame blending, then you would either re-render your 24 fps footage as 29.97 fps interlaced with 3:2 pull down added or just use it in your comp with frame blending added or use Twixtor to convert the footage to 29.97...
So I guess we need to know exactly what kind of footage you have and what you are trying to do and what you are expecting to accomplish before we can give you a process that will solve the problem.
This is along viewing, but not only will it give you the answer, it will take less time to digest than Rick's answer, and teach you everything you ever will need to know about frame rates: