I want to do research by myself and with my friends, but I don't want to apply for a degree. What are the advantages and disadvantages of doing research without studying for a degree?
Your question is a bit underspecified, but if I understand it correctly, perhaps the following will help.
Research is about learning what is true about some aspect of the world or the mind. Carrying out research is good mental training and gives you the basis both for future study and for a profession. So, it is worth doing, just for that, without regard to any results. Doing it in a group, as you suggest, is also good as you can help teach each other as you go.
On the other hand, carrying out research without professional guidance, as a beginner, can lead you to waste your time, both in understanding what is important to study and what is already known. One reason for beginning a research career as a student of a degree program is that your advisor can keep you from falling into such traps. The advisor can, most likely, point you to resources that would be difficult for you to find on your own. There are specific techniques used in research in specific fields that you may also not be aware of without access to an experienced guide.
It is also, arguably, easier to publish if you are associated with a known institution. This isn't a requirement, actually, but it has some effect.
The main advantage is that you will not be subject to publication pressures. Graduate students and other 'professional' researchers must "publish or perish". If you're doing research as a past-time, then you won't have these pressures. Of course, if you're the kind of person that is motivated by pressure, then this might be a disadvantage.
The main disadvantage is that you will not receive a degree. This may not be an issue for you personally, but not having a terminal degree (Ph.D.), will disqualify you from applying for certain kinds of government and/or foundation grants that can support your research.
My suggestion: If you are interested in research without becoming a graduate student, I strongly suggest that you take a research affiliation with a local university. Research affiliations cost the institution nothing, but will give you access to the large network of other people that are (hopefully) also interested in your research area.