When you cut, just make your main sequence in 24p and import all your 60i footage, Premiere will take care of it automatically (it will use a simple resampling, choosing the nearest frame/field from the 60i stream for each frame). Now, it should automatically detect the 60i, but if it doesn't, you can do "Interpret Footage" with a right click in Premiere and set the field order.
Now, naturally that will introduce a very slight amount of jerkiness, as the frametimes of the 60i stream dont fall exactly on the frametimes of the 24p stream, but it's your bet either way. And in most scenarios its unnoticeable.
You could slightly alleviate it with Frame Blending (blending together the two nearest frames, each's opacity weighted by how close it is), but I absolutely despise the look of it. Whenever you stop the video or look closely, you will clearly see two images overlaid over each other, it just looks terrible and screams "amateur" to me.
The only other way, if you wanted it to be ... more fluid, so to speak ... would be to use time interpolation for the framerate change. I don't recommend it though, because it is A. slow and B. introduces lots of artifacts typically. I *always* opt for the frame sampling method for that reason. I just prefer the image to look professional and clean and the interpolation artifacts really mess with that. The only case where I would consider interpolation is in a scene that has absolutely minimal and only slow motion in it, then the interpolation tends to work without noticeable problems.
If you want to try out yourself, you can just set up the project like I mentioned, drop the footage in your timeline, then right click on the clip in the timeline and there is something called "Time Interpolation" I believe, which goes to a dropdown showing you the options "Frame Sampling" (Default), "Frame Blending" and "Optical Flow". Frame Sampling is my preferred one. Frame Blending I explained above. Optical Flow is the one that does an interpolation (though there might be ways to do it outside of Premiere that are better, I am not sure).
Perhaps important note: Since your source footage is interlaced, you may consider deinterlacing outside of Premiere to create a lossless or very high-bitrate 60p stream first. Premiere does automatically deinterlace, but it doesn't do so with very advanced algorithms. I will say, it will probably look good either way, but if it's some professional project or something and you want every last bit of quality squeezed out of it, external deinterlacing is something you could do. Though the good methods also tend to be rather slow, and the intermediate files resulting from that will be rather big. Personally, I wouldn't bother, but wanted to mention it
