Do you often get the chance to write in Japanese on a computer?
It's different from handwriting, in that with a PC you input Japanese sentences using romaji. This means you are given an opportunity to check your own understanding of hiragana, katakana and kanji, and a good chance to practise them.
Although you are basically just inputting Japanese text using romaji, there are some special keyboard rules for typing certain characters, i.e. ん, -, ゃ, っ and so on. It's best to find out these rules so when you come across them you'll know what to do!
With that said, I'd like to introduce one site in particular that gives you the opportunity to learn - and practise - all these rules.
Japanese Typing Practice for Beginners (Introduction article)
Using this site you can feel a sense of achievement as you practise typing following an order which starts with single words and increases to the point where you can enter full sentences with no problems.

The site also gives you the chance to practise your dictation-type skills, as you input a passage of Japanese text you hear.
Why not try a bit of play-acting? Practise the dictation as if you were a clerk attending an important meeting held in Japanese, documenting the contents of what you hear!
There are 6 'steps' for practising typing on the site.
Once you start, please advance through these steps in order. (You can't jump straight to "Kanji typing", for instance.)
When you finish the word-level typing exercises you can move on to sentence-level typing.
In this way, you pass through various practice stages to progress to the point where you can write full sentences with a mix of kana and kanji.
At the end, as an "Assignment" you are set the task of writing a message to your own teacher.

When you wish to move on to the next step, please click on the menu found right at the bottom of the screen.
A 'kana chart' is always displayed on-screen, to help you with your typing in case you forget how a certain character is entered.
Also, for each practice stage there is a short section on "Typing methods" that gives tips for how to input certain characters using the keyboard. It's a good idea to read this first, before starting the practice.
Even during your typing practice, you can press the 'check' button to check your answer. If you got it correct, a flower-circle (the equivalent of a gold star) will appear!
If you should find it very difficult to input the romaji correctly for a certain answer, please press the 'answer' button to see an example of the right way to input the text. An example will appear in brackets [ ] in a separate small window for you to refer to.
This way, if you don't understand exactly how you should input the Japanese, instead of having to wait until the end to see your mistakes, you can check how you are doing even in the middle of your practice. This means you can progress swiftly and effectively with your studies... good huh?
Here's another website where you can practise your Japanese typing.
Online Japanese Practice (Introduction article)
The contents look like this.

Here, the hiragana and katakana practice is more simple - you look at the basic Japanese syllables and just enter them as is. This means the site might be best suited to those learners who have just started learning the kana.
For these learners I think it would be a good thing to get a real grip on the basics here before starting on the Japanese Typing Practice for Beginners practices.

Kana practice: Example 1
Rather than simply offering typing practice, the main function of this site is to give the user various tasks to check if they can conjugate verbs properly, read the numbers and understand kanji readings.
However, since the answers have to be inputted by the user (rather than chosen via multiple-choice) it also provides useful typing practice.

Kana practice: Example 2

Verb-conjugation input screen
To improve your Japanese typing, you could try using these 2 sites in the following way.
1) First, practise inputting the basic hiragana and katakana syllables with Online Japanese Practice.
2) Next, use the Japanese Typing Practice for Beginners website to progress through the different stages, beginning by inputting kana words, moving on to kanji words, then through to kanji/kana sentences and listening practice with dictation.
3) Go back to Online Japanese Practice to tackle both grammar and typing in Japanese at the same time. You can practise inputting Japanese using a PC keyboard while you answer questions on how to read numbers and construct different verb forms. At this point, understanding grammar will affect your answers.
4) Lastly, complete the final step on Japanese Typing Practice for Beginners, the "Assignment".
With the homework assignment of writing a message to your teacher, you have to exercise both your grammar and typing skills.
There are explanations about support for Japanese fonts in CosCom's (Introduction article) section on Japanese Fonts. Please take a look there too!
There is a further article in the "e! Kore" section entitled Is typing in Japanese a weakness of yours?! Solve your problems here! (Advanced version). If you're already comfortable with the basics of typing in Japanese, or if you have confidence in your Japanese ability and want to practise your kanji readings, please be sure to read that too!