If you have an older PC, and you think you need to upgrade before you get an Internet account, I have good news: you don't. With a couple of exceptions, surfing the Net is not very demanding of PC hardware. Almost anything bought within the last five years will probably work fine.
Your modem is the most important part of your PC for dialup Internet use. This is the gadget that connects your PC to the phone line. The faster the modem, the better. Get yourself a V.90 56 kbps fax modem. V.90 is the 56-kbps standard adopted in 1998 by the ITU. 56 kbps seems to be about the maximum speed possible on ordinary phone lines.
Internal V.90 modems are available for around $20. External modems have their own case and power supply, status indicators of some sort, cost a little more, and can be serial or USB. External modems apparently work better with Linux. Serial external modems sometimes need an extra serial cable (many manufacturers supply them). If you're going to use a serial external modem, check your PC's serial ports. There are two kinds of serial port connectors, DB9 and DB25, with nine pins and 25 pins respectively. DB25 serial ports were more common originally but are mostly gone from PCs now. If your PC and modem have a DB9 connector on one and DB25 on the other, you'll need an inexpensive adapter.
If you have an older or slower modem, it doesn't mean you can't get on the Net, it will just limit what you can do somewhat. "Fax modem" just means it can send and receive faxes as well as connect to computers; all modems are fax modems these days.
A dialup Internet connection may not perform well unless your modem is initialized correctly; see this Modem init strings site for initialization strings for dozens of models, plus instructions.
A majority of US Internet users have extra-cost broadband or high-speed connections now. Broadband connection types include DSL, cable modems, and satellite. Many people are still on dialup not because they can't afford broadband, but because broadband isn't available yet where they prefer to live. It's still unclear if any particular broadband system is going to become dominant for consumer accounts.
If you want to make VoIP phone calls using your computer, you will need either a PC microphone and speakers or a headset (wired or Bluetooth wireless). If you want to do desktop video conferencing you will need those plus a PC-style video camera that can be perched on top of your monitor/display. For more detailed information about VoIP and video conferencing (which used to appear on this page) see my Internet chat page in the Net software section.
You need enough hard disk space to install the Net clients you want to use, and to store any programs or other files you grab off the Net. As always, more is better. Hard disks have gotten so cheap that most people no longer have any problem with disk space, even using just their original disk. If it turns out you do need more, you can probably just plug in and format an inexpensive second drive, which can be much bigger than the first one if need be.
For your Web browser to operate decently, you need a display adapter and monitor that can operate in 800×600×256 video mode or higher. This is the basic lowest common denominator for Web graphics these days. Most systems sold in recent years should give you at least 800×600 at "true color" or 1024×768 at "high color." There's really no good reason for it, but too many Web designers these days hard-code their page widths for an 800×600 display now; it's a fad. You won't be happy "surfing the Web" using 640×480 video modes.
This is a question your new ISP is going to ask you when you sign up, and one that new users usually aren't prepared for. Here are some examples of common styles of logins, for the name "Jane Doe" and a hypothetical provider, with typical email and personal Web-page addresses.
Of course, depending on how large your provider is, your first choice for your login ID may already be taken; so you might want to have some alternative choices ready. AOL used to generate unique screen names by appending numbers to your proposed ID strings. You can do this too; you might want to pick numbers that are easy to remember, such as jdoe999@isp.com.