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As for designing, you want to make sure the site looks the same on the following browsers:
If you have a Mac handy, then test Safari as well.
Is the best idea to just style for the most popular resolution and go with that?
IMHO - Yes! (1024) Depending on the nature and content of the site, one that adjusts dynamically to screen width (width setting of 100%) can look really bad on wider screens. What was once a nice paragraph is now nothing more than a l o n g l i n e o f w o r d s.
Again, IMHO, design with a width no greater than 1024px and center the design so there is no a lot of white space on one side. A nice background image, or one appropriate to the specific content*, can be used to fill that area. Also, do not used fixed heights for your fonts, use percentages or em's so the viewer can adjust them if necessary.
* Just as an example, I did a site for a guy running for office. I made a background image of the voting form with a little filled in oval, his name next to the oval, and the text "Vote for no more than one" next to his name as it appears on the voting form. He won and I like to think my idea helped. But that is what I meant by content specific background.