は is the topic marker, not the subject marker. There is a subtle difference between the two that I don't really have enough understanding of to explain properly. Sometimes they're interchangable with very little effect on the sentence, and sometimes they're not. In this sentence, they are.
私があなたに教えます。
私はあなたに教えます。
Here the 1st version lends more emphasis to "I" in "I'll teach you." The は version sounds much more matter-of-factly.
As for your 2nd question, yes there can be multiple
topics in a single sentence. Sometimes it helps to think of は as "as for ..." or "concerning ..." instead of just noting it marks the topic. So think of that sentence as "As for me and as for today, [I] will ride." It's horrible English, but it's much closer to the literal meaning. Of course, the better translation is simply "Today I'll ride [whatever]." Here are a few more examples.
私はゲームが好きです。 ("As for me, games are liked"->"I like games")
妹は虫が怖い。 ("As for my sister, bugs are scary."->"My sister is afraid of bugs")
彼には兄弟はいない。 ("As for to him and as for siblings, they don't exist"->"He doesn't have siblings")
The following attempts at further explanation may possibly just confuse you. Read at your own risk. 
In a sense, multiple は collect together the topics into a "pool" of topics that the rest of the sentence pertains to. As a kind of graphical representation of the last example:
{"to him", "siblings"} <- ["do not exist"]
I always hated math, but do you remember the distributive property of multiplication? It's the one that says things like, 4(a+b) = 4a + 4b. It's a similar thing with multiple topics:
"do not exist"{"to him", "siblings"} = {"don't exist to him", "siblings don't exist"}
)
! i don't think of myself as smart as rather repeatedly lucky when taking math tests because half the time i don't know what i'm doing in class but then i'll end up with an A 
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