It looks like nobody has bit yet... I was hoping some of the more experienced hams would see this because I'm new to radio myself, but I can offer my input.
You might be thinking of ham/amateur radio, but I'm not entirely sure. There are a lot of radio services out there, but I'm not sure of many that let you transmit within the service's bands, other than the amateur frequencies if you study to get license to use them (and obviously the unlicensed bands).
While there are many people who use the amateur radio spectrum to do satellite bounced/relayed communications, it takes a bit of planning and lots of knowledge in how it works.
You wouldn't really use amateur radio to call a certain person in particular (I don't think most amateur satellites are connected to the phone network) but instead it is more experimental and with an intent of getting to know people around the world better through communications.
I'm part of a balloon team that uses amateur radio and GPS to remotely track our balloon with the APRS network, and it's pretty cool to see the data automatically make it to the internet through the network. But that is one of the few amateur radio applications that built a world-wide infrastructure through the internet. While there are certainly some neat things you can do, depending on what you want to do it might not be the right radio service. But there aren't really any other radio services that can be as much fun if you have an open mind and want to try things.
I just recently bought my own HT (Handy Talkie) radio tranceiver...expect to pay up to $200 for a dual band 144MHz/440MHz, which could cover most local communications (especially if your area has repeaters - ask a local club about radio and they'll explain). There are triple band ones for $300+, but they add a 220MHz frequency section that hasn't been developed as much and is quite uncommon. You might even find a used one if you ask, or might get to practice on club equipment if you get involved (and are licensed).
If you want to look into getting a Technician's license (entry level) for ham radio, look for publications from the ARRL (or your country's local Radio Relay League) since they handle the tests and paperwork with the FCC (or your country's equivilent). Cheers.
