Path to a good grad school in math during undergrad?
Here are my suggestions, having just finished a year of graduate school in math. It's therefore mostly anecdotal and should be taken lightly!

REUs: Try to do as many as you can! You get to meet other people who like math, learn new stuff, practice struggling with research, travel a bit, and get some cash to boot. They also, of course, look good on applications. 

Publications: I don't personally have any publications. I wrote a few papers during my REUs and projects, but they were only published on the REU websites. So they're not necessary to get in. However, I did have a great deal of trouble getting acceptances. Maybe a publication would have helped, but I think it's very rare for an undergraduate to actually publish a paper.

Graduate Courses: I took several of these as an undergraduate. I enjoyed them, but realize now that I should have taken them a little more seriously! I've forgotten a great deal of what I saw in them. However, I have noticed that I'm quite strong in the area I took graduate courses in compared to my peers. So they definitely give you an edge! However, don't become too obsessed with loading up with graduate courses. Three of them is quite a lot of work, if you give them justice. Since most graduate courses are graded very lightly, you can make high marks in them without putting forth as much effort as you would in an undergraduate course! (At least, this was how it worked at my undergraduate institution.)

That said, keep in mind that some of your time in college should be spent having fun, too. Don't become a math robot just yet! You have time for that in grad school. :)

Reading Textbooks: The fact you're already reading the "core" undergraduate books before even entering the university puts you far, far ahead of the curve. Many people won't learn those things until sophomore or junior year. I certainly didn't. Make sure you're doing the bulk of the exercises in those books, especially Rudin. Try to prove statements you come across without looking at their proofs. I feel that this is where most of the learning happens. You can easily read things and not understand them, so just watch out! Other than that, finish those books and then you should be set to take the advanced undergraduate/first year graduate courses at your institution.

The Math Subject GRE: I hate this thing and did very poorly on it. You'll blow the math portion of the general GRE out of the water. It's easy stuff for any math major. However, if you don't spend a little time reviewing, you can really mess up the subject test, since it's timed and covers things you might not have thought about for several years. The topics are almost completely disjoint from what a student taking graduate courses has been doing. Look at a few practice tests, identify what sorts of questions are asked, and train yourself to quickly answer such questions. Speed is key. You don't want the subject GRE to be a weak point on your application, especially since it's an easily prevented weak point.

So, do REUs, take graduate courses, don't sweat not having publications (but if you can get some do), and study for the subject GRE!