I have an emachines T3624 that I bought brand new from Best Buy in 2005. It has Windows98 XP on it of course.
Just to be clear, Windows 98 and Windows XP are two completely different Operating Systems. Your system almost certainly has Windows XP; Systems weren't shipped with Windows 98 much after 2002 or so.
(1) to sell a few postcards on ebay, so I have a printer/scanner too.
(2) a little emailing
(3) looking up info/checking the news
(4) store a few photos and some music
(5) check my bank account info
I do not play games online and even though I have AT&T DSL internet service, I find that I can seldom watch a video without waiting for it to keep loading up (buffering), so I give up.
I know that Windows98 XP is now all but dead. Money is really tight at my house (and getting tighter!) 
However, if 98XP makes my pc websurfing dangerous b/c of security glitches, I'm assuming that I need to get another pc and soon! Sigh... 
For your purposes, I don't think a new PC should be necessary. If security is a concern, you
could do all of those things with a Linux machine.(EDIT: I should clarify: Linux is a Operating System that you could use instead of Windows XP, not an actual machine) The trade-off, of course, is an investment of your time to learn that system, so it depends if that is worth it.
I will have to buy a used one, but want to do so through a reputable dealer like walmart where I can at least get some kind of warranty!
I need to know if most desktop pcs still have ports for a telephone line as it looks like I will be having to switch to a dial-up ISP like netzero or such to save money. 
Computers have never had these ports "built in". These ports- the Telephone jacks, that is- are part of a device called a 'Modem'. For some time, computers came standard with a modem. Even my laptop (~2008 or so) has a modem connection.
They have not, however, typically been included in desktop machines for some time, instead being replaced for ethernet on a add-on card or more recently the system board. Modems are however relatively inexpensive. Additionally, if your current machine has a Modem it might be possible to remove the modem card and install it in the new system.
When I was looking on walmart's site about buying a used pc, they mentioned trading in your old pc and getting money for it. What can I do to get as much of my personal info off of my pc as possible before doing something like that? I don't want my pc to go in a landfill either afterall!
Though I Suppose this depends on the person I personally avoid throwing out any of my PCs. At the very worst I end up storing them in a closet. I also suspect that Wal-mart would probably be using the trade-in thing to get old computers which they then refurbish and resell, probably for more than they bought it from you. If you still want to you can use a tool called DBAN (Darik's Boot and Nuke) Which you can burn to a disc. It will completely clear your hard drive of any data. Another alternative- and because I tend to be a preserver- is to simply remove the hard drive entirely. Be sure you have all the data you want from the drive, especially in the first case.
And what is the earliest Windows OS I should be willing to accept on a pc? I know that 98 XP is OUT!
Vista. You won't be able to buy a new PC with anything older than Vista anyway. You'll usually end up with 7 or 8.
My 9 year old pc is still working. I've never had a problem with it. I'm using it as I type this now! But I guess that I have to face reality since 98 XP is "gone".
IMO, if the PC still works, and it is not powerful enough for Vista or Later to run well (2005 means no I would expect). I do not see a problem with sticking to Windows XP in this particular case. It is less than ideal but if money is truly tight you should not feel that you are in a position where you are "forced" to buy a new system. An alternative worth exploring before you decide to buy a new PC- especially if money is tantamount- would be to try a Linux distribution and see if that can fit your needs. As I mentioned this would still "cost you" in terms of time taken to learn it, but it would still leave you with a heavier wallet (or more digits in your bank account), so could be worth exploring before spending money.
If you decide to try the Linux route I'm sure many members of this forum who are familiar with it could help guide you through it to help you determine if it would fit your requirements. And in regards to the parts, it is fairly easy- you compared it to removing a transmission, but is quite a bit simpler than that thankfully. And again I'm sure if you decide that could be a option we would be more than happy to try to help you through the process of taking say the Modem card from your current system and bringing it forward to a new system, if you decide to go that route. It could save you 10-20 dollars for the USB Modem Camerongray mentioned, so could be worth it.
