Is there a way to save your phone's address book?

At the Sprint repair shop, they sucked the old address book out of my broken phone and poured it into my new phone.
Is this something civilians can do?
I’d just like to back it up from time to time.

It depends on the phone. You can buy data cables and software which are supposed to do this. Search for your phone model and data cable. I have not tried this myself but I have heard that people have mixed results with this.

http://www.wirelessground.com/dacamosanolg.html might have such things

I have done this with an LG VX4400 using a third-party cable and open-source software. I have done it with a Motorola E815 using a Motorola cable, and Motorola PhoneTools software made by BVRP. (BVRP makes tons of phone-related software for mobile phones; they also made the answering/fax software that shipped on my Dell Dimension 2400.) Both programs also allowed transfer of multimedia data like images and ring tones (however, the Motorola required some hacking to do so).

Both phones were under Verizon Wireless. Verizon also sells the software and cables to do this for many, but not all, of their phones.

I haven’t done it directly from one phone to another, but I do pull the info off of my phone (Sprint - Sanyo MM-7400) and store it on my home and work pc. And I’ve made changes to the data on the pc and transferred it to my phone (easier than messing with the phones keypad, for me at least.)

I bought a data cable on ebay for less than $10. I also use an open source program that works well enough. (BitPim)

I just moved the SIM card. Like it was a removable hard drive.

A SIM card is for GSM only. Not that many GSM netoworks in the States; the OP referred to Sprint, which IIRC is CDMA and does not offer GSM. (Sprint used to be a partner in Sprint Spectrum, a GSM network in the DC area, but the sold out and eventually T-Mobile picked up the network. Maybe you’re a T-Mobile customer.)