There are many ways. First off, are you in a Mechanical Engineering program in college? If so, there are many groups and activities you can get involved in. I went to Virginia Tech from 2002-2007 and I had options like Formula SAE (which is a project to develop a small Formula-style race car. Here is the full list from Virginia Tech: .
Even if you are not in college there are still ways to get involved. I live in Charlotte, NC and there are places here called “makerspaces” where you can become a member for a small fee and then have access to all kinds of great equipment like 3D printers, laser cutters, CNC machines, etc. Here is the link to “Hackerspace”, which I recently visited:
My advice is to pick a problem you currently have, and then start to work on a solution. For example, I had a simple issue of my home coat hanger leaning over. I made my own mechanical design project by solving this issue. You can read about it on my blog post:
Now obviously you will need some equipment to get started but if you are truly interested in mechanical design, it is worth it in my opinion. For my coat hanger project, I used calipers, CAD software (specifically, Solidworks), and my 3D printer.