Generally, MP4 is better for compatibility, whereas AVCHD provides the better quality. MP4 (which by the way can mean a million different things, in this case it probably refers to MPEG-4 Part 14) files will be compatible with most devices and media players as is, and will be smaller in size compared to AVCHD. This is the option I'd recommend if you wanted to film your vacation and show the recordings as is, without further editing.
If you want the best quality, go with AVCHD. Files will be bigger in size and won't play back on most devices, but you can import them into Premiere Pro and export your edited video to every format you want anyway. Chosing the AVCHD will also open up other recording options on your camera. According to the manual, the highest bitrate avaiable for MP4 is 35Mbps. For AVCHD, the highest possible bitrate should be even higher, though I couldn't find an exact value.
Can Adobe Premiere CC work equally well with both file types?
Yes. MP4 will record, well
.mp4 files. AVHCD will probably come as
.m2ts files, both of which Premiere Pro can handle.
Which file type will give better image quality?
AVCHD with the highest recording quality settings. Keep in mind that this will fill up your SD card pretty quickly, according to the manual at about 4 GB per 20 minutes of recording. If this is a problem for you, MP4 might be the better solution, as it offers the better quality/filesize-ratio.
Does a single AVCHD clip really comprise multiple files that I'll have to import?
Not really sure about that. Normally, it should produce one container file every time you press record. Many cameras will split the file at 4 GB or stop the recording when a continuous recording reaches that size. Since you already bought the camera, just try out both options and see what files you get. But Premiere pro should be able to handle whatever you throw at it.