As with kana, both reading and writing of Kanji is very important. In today's computer-based world, of course, writing may seem less important, as computers can translate kana directly to Kanji, but writing is important nonetheless, and can also improve your reading abilities. Think about it: if you can write a certain character, then you can certainly read it! And also, due to the frequency with which Kanji are written in calligraphic styles, understanding how they are written can be crucial to being able to read them. Like kana, there is a specific stroke order for each Kanji. There are minor variations between regions and handwriting styles, but the vast majority of the time, the standard stroke order is a must, especially for a foreigner learning the language.
So, which Kanji do you learn first? If you have a textbook, it may have a set of Kanji for each lesson, perhaps relevant to the vocabulary or dialog. If not, the best Kanji to learn first are the most common, as they will be the most useful. Numbers, days of the week, and your favorite verbs are good starting points, as they will be used multiple times per day. If you are self-taught or trying to work ahead in your class, this is a great way to begin, because it will be simple. If you write a lot of sentences with the verb taberu (to eat), then learn the Kanji for the ta- and you will be set. If you have class assignments each Friday, learn the Kanji for the kin in kinyoubi and you can impress your teacher. The key when starting out is to find the words you use the most and learn those Kanji first, as this will allow you to start using Kanji immediately. Once you have learning a particular Kanji, try to use it every single time you can, to increase your familiarity with it. The native Japanese typing systems on both Windows and Mac computers will automatically convert kana to Kanji, though you must be careful to pick the correct one on occasion, so even if you don't do a lot of writing, you can still practice your Kanji often.