Some Kanji I do know right off. My problem is actually writing them. I don't know the correct direction for most of these characters presented to me.

Oh, and for words like "today" and "this week", that's easy to remember because of the first Kanji there. I know that it's "ima" which means "now", so I remember that it's "present tense", for the most part. I also know the kanji for "day". It was one of the first ones I recognized.

What do actual Japanese Elementary School students start off with? Is there a good book on how to write Kanji? I know repetition is the key, but I'd still like to start at the beginning, to be completely honest.

i have the tuttle books with 250 kanji in each and they are pretty good, they give compounds that start off easy and get more difficult as you learn more. they show stroke order and sometimes some extra information about the characters origin etc. I didn't use them after about the first 30 or so characters just because i went ahead with my method i stated earlier.. stroke order is really dead easy to pick up.. learn the radicals and their stroke order and you pretty much will know the stroke order for every character out there.