Ok, so let's review this and try to figure it out.
First off, the reason you want to do this is that a wired connection is less likely to result in corrupted data being passed back and forth, and when it comes to transferring firmware onto the router, it's CRITICAL that there be no disruption, as you get ONE SHOT and good, bad or indifferent that's the data it encodes into your router. Screw it up, and you've just bricked the thing.
The default address for the router's setup pages is the IP address of the router itself. On Linksys brand routers (as well as others, by the way) the default IP address is 192.168.1.1. From your point of view as the owner/user of this thing, this might seem like an arbitrary address to know, however the thing to understand is that there are several "private IP" address ranges as defined by the standards which control network and Internet addressing schemes, and the 192.168.x.x range is one of those. It's probably the most common of them, especially in the consumer product space.
Anyhow, don't mean to go off on a tangent, but I thought you might benefit from understanding that. And in any event, the default address that Linksys uses is ".1.1", so therefore logically since they use the "192.168" range, the address for the router has to be 192.168.1.1. Now, it happens to be true that this is changeable by you; however that's a subject for a whole different discussion and really isn't germain to this one here.
So, getting back on track... You connect one of the ports (not the WAN port, of course!) up to whichever computer you want to use. This is a Mac message board, and you have said you own a Mac, so I assumed you'd want to do this on your Mac. However, if you would rather do this on a Windows PC, that's fine, too. For our purposes, whatever platform and browser you use is immaterial. Just accomplish getting this router hooked up via a wired connection to some one computer for purposes of following my earlier directions.
Also, and I should probably have mentioned this before, but make sure nobody else is trying to use this router while you're doing this work, otherwise you risk causing all sorts of problems, nonetheleast of which is bricking the router. That's bad; don't do that.
As for not being able to get to where you want on the Mac but being able to do so on your PC, that is strange. And no, I have no idea why it would do that. As I said earlier, do this via your PC if you need to.
I hope this is of some help. Don't hesitate to ask further questions -- you won't continue to learn and grow if you stop asking.
Thank you so much, sir, for taking the time to help me out. I finally figured it out and was able to get everything up and running after I restarted my MacBook...
Thanks again! Much appreciated.