There aren't any kanji that you should learn before writing in hiragana. You know, when Japanese children start learning to read and write, they start with hiragana and write everything with them. After learning to do that, they continue to kanji. (I'm not sure when they learn katakana and romaji.)
Like you said, kanji is often used for the nouns, but also for the stems of adjectives and verbs. Usually you can find the stem if you try inflecting(?) the word, ex. samui, samukatta... samu- is the stem, and the rest is written in hiragana: 寒い, 寒かった. Kanji is used more seldom for particles, and for onomatopoetic words or the particles like "wo" or "ni" only in ancient Japanese. I use kanji because it kind of cuts the text into nice parts, which makes it easier to read. (Usually you don't use spaces in Japanese, but you might want to if you write long sentences in hiragana.)
My basic rule nowadays is "don't use kanji you don't know." If there are spelling or grammar mistakes in the sentence, the input method editor might not convert it correctly or if you convert your text word by word, the result might be just wrong. However, if you've just looked up a word in a dictionary, the kanji version might be better because there are lots of rare words that share the reading with some other word. Still, dictionaries have some kanji that isn't used very often. Natives might comment them with "we don't use the kanji for that" or "I had to look up that in my kanji dictionary. I had never seen it before!"
As for the first part of your question, there are some kanji that are good to recognize even if you're a beginner:
日本 - Japan, 日本語 - Japanese (language), 日本人 - Japanese person/people
(The same kanji are used for other country names, too.)
