Much like being a first generation college student, you can be a first generation Greek. No one in your family has had any experience with Greek life and if they have, it was as a GDI.
I was a first generation Greek myself. At first, my family was a little taken aback when I announced that I was going to go through sorority recruitment. A lot of people told me that I didn't seem like the sorority type while others said it might be good for me to experience this.
One main thing to keep in mind when you first consider going through recruitment is that everyone will have an opinion. Whether they think Greek life is great or needs to be wiped from the face of the Earth, everyone will have an opinion and they will voice it. If your family is starkly against Greek life I suggest taking them to informational meetings, meeting with the Greek life coordinator at your campus or exploring the events and activities that have been hosted by the Greek organizations at your school and then showing them your findings.
Another thing to think about and talk to your guardians or parents about is the financial aspect of Greek life. Chapters are normally very good about getting PNM's as much information about the financial requirement of their organization as possible. They do not want to convince you that you can afford it only to have you join and later figure out that you can't. Most girls who figure out that they can't afford their sorority tend to drop rather than try to work out their financial situation with the chapter's treasurer. If you join and think that you can no longer afford your sorority please do talk to your treasurer.
Some parents tend to worry about the time that Greek life can call for, they believe their children should be studying or devoting their time to other things. Lots of PNM's worry that if they are already busy with school, work, relationships, and family that they cannot manage to integrate a new organization into their already busy schedule. Adding anything to a busy schedule is doable with time management. One of the best things that a college student can have is a planner. Between work, school, and anything else you choose to be involved in a planner is a necessity. I thought I could get through Greek life, family time, two jobs, school and a relationship without having a calendar to keep everything straight. I was wrong. Now I have two forms of tracking my events, a planner I can use during chapter meetings and a calendar on my iPad.
The best thing you can do when your family is concerned about your interest in Greek life is to explain and share. If you can sway your family into letting you try then you can better show how you feel about it and the benefits it brings you.