How to backup and restore only applications on Mac

Okay so here is my problem...I transferred everything from my old Mac (250 GB) to my new Mac (125 GB), and in the process, all the "other" was also transferred. 82/125 GB on my hard drive is "other." I transferred all my files onto a second external hard drive (not my time machine one) and basically there is nothing on my Mac except for my applications (~15GB) and stuff in iTunes (don't care about that) and then in the documents folder (~6 GB) there are things from MS Office that I may or may not care about depending on if they are important lol. So what I want to do is basically just backup the applications onto time machine and maybe the documents folder. So if I exclude everything except those two, it's showing up as 97 GB for a full backup (~15 GB + ~6 GB is not equal to 97 GB). I'm sure there are system settings being backed up too, but still it shouldn't equal to 97 GB. So will all the "other" be gone. I understand that any Mac that is being used will accumulate "other." But I don't want the "other" from my old Mac to take up all the space on my new one. I hope this makes sense. But anyways, I have a bunch of applications that aren't from the App store (E.g. MS Office and a few others) that I don't want to have to pay for again, I just want to back them up and then basically start my Mac as a new laptop with nothing on it and then put all my applications back on it so that there is really only applications on the computer.