A lot depends on what you want to do with the files long-term. Like photographs, if you process themand then save as a different type of file you are likely to lose data which is then lost forever. That is how Jpeg pictures and MP3 files manage to be smaller.
The small files are easier to store, and with audio, this means that you can fit a lot more tracks on your portable music player. The track will usually sound fine, but the problem comes if you try to re-process the already degraded file. For instance if you convert Real Audio to MP3, each format throws away different bits of information, so when both conversions removed data from your original, you may start to notice the missing sound quality.

Your decision depends on what your original file format is. If you have started with MP3 file downloads, you will not improve the quality by converting to something else, whereas, if you have an original digital recording in Wav format, most processing is likely to permanently reduce the theoretical sound quality, although it may not be noticeable for listening.

What I tend to do where possible is keep an archive copy of the full quality track, saved somewhere sensible. I can then make a copy from this "Master" which may be saved as some other format convenient for listening to. Most portable players limit you to Wav, Wma and MP3, so if you are using one of those, it becomes irrelevant whether Ogg Vorbis is a better quality compressed file format compared with MP3, the player will not play it.

As for the tracks saved on your new hard drive. Decide whether your intention is to have them there for listening, or whether it is as an archive for long term storage and possible later conversion to "listening" tracks. If it is storage, you might want to save in a compressed format which will keep the original data intact. Something such as a "Zip" archive will take up less storage on your disc, but you would have to "extract" files before you can listen to them. For music files, there is a format I think it is "The Monkey's Audio" which is specifically designed to keep all the original data from a Wav file, but compressed into a much smaller file. As far as I am aware, the file doesn't play as music, but when needed, can be converted back to the original uncompressed version.

Quack!
Geoff the Duck.