I know how to read hiragana and katakana, but know nothing else. A lot of people I know want me to write their names in Japanese. Is it right to write them in any system, or are there specific rules?
Names of Japanese people have a specific spelling that may be in hiragana, katakana, kanji, or any mix thereof. You should spell their name as it is spelled, e.g. 「田中けい子」 (TANAKA Keiko).
Names of foreigners are written in katakana, unless they have decided on the equivalent kanji they want to use. Or if they ask you to choose the kanji then do so paying attention not only to sound but also meaning when selecting them.
You should keep in mind writing non-japanese names with kanji COULD be seen as somewhat not in very good taste. While it's there's no universal rule about it, since some people might see it as ranging from childish to culturally-insensitive, I'd keep using exclusively katakana for foreign names.
Straying a little from your question, it's generally a nice idea to write names in whatever script and with whatever spelling they originally are whenever possible.
Write Japanese names the way how the person who has that name writes it. If one wants you to write her name as 玲奈 or 麗奈, write it that way. If she doesn't write it with kanji, then write れな. If she prefers katakana, then write レナ. A name might use more than one writing system, such as はる香 or マナ美 (I made those up, but very possibly exists somewhere). Using the wrong writing system for native Japanese names is just, plain dong it wrong.
So, if you don't know kanji, unfortunately you do have to learn them before you could correctly write Japanese names, since practically all Japanese names have kanji in them.
For non-Japanese people, especially those with Western names, the usual way is to transliterate their names into katakana. If you only know katakana, I suppose you could write their names correctly.
However, if a person has a name written in Chinese characters in his/her native language (i.e. a person with Chinese or Korean name), write it with kanji. The exception is when the person's name contains rare and uncommon kanji (especially when typing); in this case one has to resort to using katakana - simply because these characters might not be typable.