Basically you use hiragana in Japanese words, grammatic extensions like verb endings or particles. For example in the verb tabemashita(食べました) tabe(食べ) is the stem usually written with kanji included and mashita(ました) is the verb ending which is written in hiragana. The particles は, が, か, を and so on are all written in hiragana. Some words like 若しかして could be written using kanji, but in common Japanese they're also written in hiragana only (もしかして).
On the other hand you use katakana only for foreign names, words OR Japanese words you want to highlight.
ホワイトデー White Day
チョコレート Chocolate
In the case of "タイヨウのうた" they didn't want to write it using kanji (太陽の歌), they wanted to make it look more special, so they only used kana(hiragana + katakana). I think they used katakana to write taiyou to highlight it even more.
If you've memorized the kana the next step would be learning kanji and/or learning grammar. But you'll notice very soon that you just can't avoid learning kanji.
Here's a nice site with good grammar lessons:
http://www.learn-japanese.info/indexg.htmlBut studying Japanese using free online lessons is just not enough imo. You should get yourself the Genki books I + II. They're the best imo.
Here's an amazon link:
http://www.amazon.co.jp/GENKI-Integrated-E...I/dp/4789009637