How to use these verb forms correctly

I have written my explanations specifically for people whose first language is Japanese. However, you might want to start with this "Verb Tense Tutorial"from Englishpage.com, or at least with some of the tests on the page, to see where your weak points lie.

1. The present continuous tense (is -ing)

This is similar to Japanese ーている but not exactly the same.
a) It is the same because it is used for actions that are going on at the time of speaking/writing (but will stop later), and for repeated actions (that will stop later). For example:

He is getting ready now, so please wait for him.
They are speaking Japanese even though the teacher has told them to use only English.
Be quiet! I am trying to concentrate.
He is wearing the sunglasses that I gave him.
We are selling these sausages cheaply because there is a special offer this week.
They are shouting Beckham's name over and over again.
He is always ringing me late at night. (Here there is a sense that the speaker is complaining.)

In this use, it is often found with words/phrases such as 'at the moment', 'now', 'still'.

She is still eating her lunch.
Everything is going well at the moment.

b) It is different from Japanese ーている because it is NOT used for habitual actions (that will continue to occur):

I come to Hiyoshi every day.
I normally get up at 5.30 and go to bed at 12 midnight.
He speaks French with a Canadian accent.
She wears contact lenses.
We sell everything at cheaper prices than other shops.
He works for a British company.

c) Also note the differences from Japanese ーている in the case of the following verbs:

知る	インターネットを上手につかえるようになったら、最新の情報を知ることができます。	If you learn how to use the Internet well, you can find out the latest information.
	新聞で知りました。	I learnt it from the newspaper.
知っている	今頃の学生はインターネットのことはよく知っているけれども、本のことはあまり知らない。	Students nowadays know a lot about the Internet, but not much about books.
着る	もう時間におくれているので、コートをはやく着てください。	Please put your coat on quickly as we are already late.
	リサイクルショップで見つけた服を着るのが好き。	I like to wear clothes that I've found in second-hand shops.
着ている	リサイクルショップで見つけた’６０年代のミニスカートを着ている。	I'm wearing a sixties' miniskirt that I found in a second-hand shop.
閉まる	ドアがしまります。	The doors are closing/about to close.
閉まっている	ドアがしまっています。	The doors are closed.

Similar Japanese verbs include:

持つ	hold/take	持っている	be holding/have/own
はく	put on/wear	はいている	be wearing/wear
死ぬ	die	死んでいる	be dead

2. Past tenses

In English, the viewpoint of the speaker or writer is very important in deciding the tense.

If the focus is on the fact that the action was completed at some time in the past, use the "simple present".
I finished the paper an hour ago.
He first visited the United States last summer.
An explosion occurred in which several people were severely injured.

Your viewpoint is in the past, at the time when the action occurred.

However, if the focus is on the situation now, rather than on the time when the action was completed, use the "present in the past"(also known as the present perfect or the past present). This is similar to the use of 「もう」in Japanese.
I have (already) finished the paper. (The fact that the paper is ready is more important than when you finished it)
He has visited the United States at least once. (It does not matter when he visited the United States.)
An explosion has just occurred in which several people have been severely injured. (What is important is the fact that we are still dealing with the effects of the explosion.)

Here are some short dialogues:
A: Were you able to finish your homework last night? B: Well, I finished everything except my English rewrite. I am going to ask the teacher for some more time.

A: Have you finished your homework? Everyone is waiting to leave. B: Yes, I have just answered the last question.

A: Does he know much about the United States? B: Well, he has visited it several times.
A: When did he last go there? B: He spent about two weeks in New York last year.

For a more detailed explanation that includes a section about 'had' + '-ed' (generally known as the 'past perfect' or 'pluperfect') see this page from Englishpage.com. There are links to exercises at the end. At the left of the page there are links to explanations of other tenses

'has' + '-ed' is normally known as the 'present perfect' or 'present in the past', or possibly 'past present'.

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a) In Japanese, it is possible to use both past and present tenses when talking about the same past event. In English, however, you must not change your viewpoint.

3. "would", "could" etc.
I will write something about this eventurally. In the meantime, try these pages from the Capital Community College Foundation on modal auxiliaries and conditionals. (There are links to quizzes as well.) There is also the more thorough "Conditional Tutorial" featured on Englishpage.com.