Ok, I'll come out of hiding for this one....
I'm shocked at the number of people who claim to still be using 800x600. I remember my first computer had a PCI video card with only 2.25mb (STB Lightspeed 128 if I recall, those were the days) and even with that little bit of memory it was able to drive a 17" monitor to resolutions higher than the 1024x768 that I ran on it. Seriously, any video card produced since the late 90's is capable. If your computer is older than that, I apologize and point you in the direction of Dell as I can only assume you're not into building your own.
If you computer is newer than that, the only plausible reason to be running 800x600 is the monitor. For starters, if you're using an LCD, any LCD, you should be running whatever resolution it is designed to run at. For 15" screens that's 1024x768 and for 17-19" screens that's 1280x1024. While the LCD's will run other resolutions, doing so results in far poorer image quality than is ideal. As for CRT monitors, anything 17" or larger should NOT be running 800x600. On a 15" CRT, I'll agree that it depends greatly on the image quality of the monitor and video card as to what resolution you run on it. In the event that you own a 15" monitor, maybe it's time to upgrade. While I realize the monitor you have may work just fine, that doesn't mean a newer one won't be a profound improvement. Flat-CRT monitors are a dime a dozen anymore with prices under $200 for 17 and 19in sizes. While they're bigger than LCD's, they don't cost anywhere near as much and produce stellar image quality. (Little known fact: A high quality HDTV Tube TV (like a Sony) will still out-perform DLP, LCD and Plasma TV's). If you have the extra cash, LCD's will save you some desk space.
If you need recommendations, try these:
As for the website design question, I've seen what's possible in the larger size and it truly is beautiful (more information, less clutter), but I'm somewhat spoiled (1600x1200 on a 20" LCD) so I'll withhold my vote for now.