"Computer Guy" will probably tell you that the best way to upgrade is to buy a whole new computer. He is not nessecarily just telling you that to make a sale! Most computers can have such things upgrades as CD drives (to a DVD or a burner), hard drives (to a larger one), and ram (to more ram). But some computers are just too old to accept the current components, and you may find that a new bare-bones system is more cost-effective. Your best upgrades for the money, if you keep your current computer, will be to get a DVD burner and possibly a larger hard drive.
Now, when you go to WDW next monday, I assume you will be taking your digital camera with you, right? Okay, then you can start your CD burning experience right there - the camera shops in each of the 4 theme parks will burn your pics from your memory card to a CD for you. Cost is $12.99 per CD, limit 120 pics per CD. The camera shops in MK and MGM also have sel-service kiosks to burn your pics to CD, which can be quicker than leaving your memory card and coming back later.
If you have a rental car, all of the local Walgreens stores in the WDW area also burn your pics from memory card to CD. Cost here is only $2.99 per CD, for as many pics as the CD will hold. The Walgreens photo labs are also open 24 hours.
After your DH gets a burner on that computer, make sure to burn all of your digital pics, old and new, to a CD or DVD. In fact, burn an extra copy and keep the extra copy at work instead of at home. This is called "off-site backup" in computer circles, and is the best way to safeguard important computer files, because fire, flood, or burglary are all unlikely to strike both home and work locations.