Is it better to raise the volume or to use a compressor to do so? (In terms of clipping)
Because when I use a compressor ofc I can make the sound louder, but by doing so I squash the sound, right? Is it better for the sound quality to sound the volume or doesn't it make a difference?
It's important to remember that a compressor does not make a sound louder, it reduces dynamic range. By bringing down the peak values, you give yourself more headroom to bring up the overall signal level (so the quieter parts can be louder).
It is that "make up gain" which makes a sound louder, not the act of compression. The choice between using compression and volume should be dictated by the needs of the context and how important transient information is to the signal you're working with. Something you might want to try experimenting with is parallel compression.
You duplicate your signal, by either duplicating the track or using an aux send, and apply compression with make up gain to that second signal. Then you have simple fader control over the compressed signal that you add in to your unprocessed original.
It's not the right choice in every context, but it's another option...and it can help preserve the transients while giving you the ability to increase the perceived level of the sound source.
EDIT: I should also note that compression does not necessarily act on the only the peaks of an audio file. It's all based on attack and release time. It can actually be used to enhance transients, and even affect harmonic content within your signal. There's a lot more to compression that just crushing the dynamic range to make things louder, and it would be wise to start exploring those.
I'm honestly not sure what you are asking here. Compression does not increase the "volume" of a signal, it decreases it. Compression makes a quiet portion of the sounds louder relative to a louder portion by reducing the signal strength when the signal strength is high.
Often a gain is applied after compression to keep the signal strength up, but this is no different from any other gain.
Practically speaking they are completely different things. You adjust gains (volume) in order to ensure a signal does not overload. You use compression in order to reduce the difference between the lowest value (quietest time) on a signal and the highest (loudest time) of the same signal.
For example, if you had a signal that was entirely composed of quiet whispers, you would turn up the gain such that the whispers were using the full available signal range (making them louder). If, on the other hand, you had a signal that contained someone whispering for a while and then shouting, you would want to apply compression so that while the person is whispering, you can still hear what they are saying, but when they shout, it doesn't get too loud.
As a qualitative approach to this question, I think you should look at it from a limiting point of view if you're trying to avoid clipping. This will allow you to keep the raw signal uncompressed and also control the peaking. I agree with @Shaun Farley about parallel compression, you should check out what it can do. Bare in mind its a little bit of black magic and also very tempting to always use it.
One interesting thing about compression is that it signals, when compressed, hit the speaker cone harder (as in intensity). A very good (published) mastering engineer told me that over Skype a few years back and it always stuck in my mind during mix down.
Ultimately if you are compressing a single instrument it can seem like the compression only increases your volume. Compression really does it's magic within a mix of other instruments. It is key in defining which instruments or vocals are up front (without having to raise volume) or when you have two or more instruments that are of similar tonalities, compression can make it much easier to have them all heard without cancelling each other out. Of course EQ comes into play there too but compression is an art form that totally aids in creating dimension once you understand it. And even then you will always discover interesting new ways to apply it. You might also look into multi-compressors. I have found those to be quite useful as they allow you to compress at a particular frequency and bandwidth. It's as if an EQ and a compressor had a baby!!
