Let's have a look at how we establish past tense with adjectives.

For '-い' adjectives:
1. Noun は Adjective format
The 'い' ending of the adjective is changed to 'かった'.
例えば： 今日の太陽は眩しかったです。 Today's sun was dazzling.
           彼の笑顔は優しかったです。 His smile was nice.
Again, to change from a polite phrase to an informal one, the 'です' is dropped.
例えば： 昨日は嬉しかった。 Yesterday, I was happy.
           本は黒かった。 The book was black.

2. Adjective + Noun format
'-い' adjectives remain unchanged in this format.
例えば： 暑い日でした。 It was a hot day.
           青いペンでした。 It was a blue pen.
To rephrase into informal, 'でした' is replaced by 'かった'.
例えば： 寒いアイスクリームかった。 It was a cold ice cream.
           嬉しい犬かった。 It was a happy dog.

For '-な' adjectives:
1. Noun は Adjective format.
Like previously, the adjective doesn't change, the ending (でした) reflects the past tense.
例えば： あの子は静かでした。 The child was quiet.
           世界は不思議でした。 The world was mysterious.
Again, as with '-い' adjectives, to make sentences to informal speech, the 'でした' is replaced by 'だった'.
例えば： 哲は変だった。 Tetsu was strange.
           その園は綺麗だった。 That park was pretty.

2. Adjective + Noun format
As with present tense, 'な' is attached to the end of the adjective preceding the noun.
例えば： 便利なアイテムでした。 It was a convenient item.
           不細工な子でした。　I was a clumsy child.
As before, 'だった' is added onto the end to create informality.
例えば： 変な人だった。　He was a strange person.
           安全な車だった。　It was a safe car.

Adjectives in Japanese come in two forms - adjectives that end in 'い', and adjectives that have a 'な' added. Like verbs, adjectives have their endings changed to signify present/future and past tenses as well as negativity.
Let's start with the present/future tense.

For '-い' adjectives:
1. Noun は Adjective format
Here, the adjective is used as it is.
例えば： 北海度は寒いです。 Hokkaido is cold.
           マイアミは暑いです。 Miami is hot.
The above sentences are in a polite form. To be informal, the 'です' at the end of the sentence is not required.
例えば： 牛丼は美味しい。 Gyu-don is yummy.
           この本は青い。 This book is blue.

2. Adjective + Noun format
For '-い' adjectives, the adjective remains unchanged.
例えば： 赤いペンです。 It is a red pen.
           悲しい歌です。 it is a sad song.
To change to informal form, the 'です' is changed to 'だ'.
例えば： 美しい女だ。 A beautiful girl.
           甘いキャンディだ。 The candy is sweet.

For '-な' adjectives:
1. Noun は Adjective format.
In this instance, the adjective is used as it is.
例えば： 世界は不思議です。 The world is a mystery.
            その木は綺麗です。 That tree is pretty.
To change to informal with a '-な' adjective, the 'です' ending is changed to 'だ'.
例えば： この子は元気だ。 This child is healthy.
           先生は変だ。 Teacher is weird.

2. Adjective + Noun format.
For '-な' adjectives, a 'な' is added onto the end when proceeding a noun.
例えば： 便利な商店です。 It is a convenient store.
           素敵なゲームです。 A great game.
Again, to change it to informal usage, the 'です' is replaced by 'だ'.
例えば： 人気な漫画だ。 It is a popular manga.
            静かな所だ。 A quiet place.

Besides depicting past and present tenses, verb endings can signify other things too! One thing verb endings can depict is the want to perform an action.

Want to do something -たい
Adding the ending 'たい' signifies the want to do something. Keep in mind that this is different to wanting something!
1. Strong verbs. To put '-たい' at the end of strong verbs, the 'う' sound is changed to an 'い' sound.
例えば： 買う = 買いたい
           私はハンドバグを買いたい。  I want to buy a handbag.

2. Weak verbs. Simply drop the 'る' sound and add '-たい'.
例えば： 食べる = 食べたい
            ミッキさんは天婦羅を食べたい。 Mikki wants to eat tempura.

3. Irregulars.
例えば： する = したい
            私は日本語の宿題をしたい。  I want to do Japanese homework.
To make the sentences in polite form, simply add '-です' to the end of the sentence.

How about if we don't want to do something?

Don't want to do something -たくない
This verb ending is also very useful!
1. Strong verbs. Like before, the 'う' sound is changed to an 'る' sound before '-たくない' is added.
例えば： 遊ぶ = 遊びたくない
            健太君は遊びたくないです。 Kenta doesn't want to play.

2. Weak verbs. Again, as before, the 'る' sound is dropped before '-たくない' is added.
例えば： 教える = 教えたくない
            私は会計を教えたくない。 I don't want to teach Accounting.

3. Irregulars
例えば： する = したくない
            有子は数学をしたくないです。 Yuuko doesn't want to do Maths homework.

Unlike in English, Japanese do not have words such as "not" to signify a negative in a sentence. This is achieved in Japanese through changing the ends of verbs. Again, there is a polite verb ending, and an informal verb ending.

Let's have a look at present/future tense first...

-ません form (polite)
Adding on '-ません' to the end of a verb makes it negative, in the present/future tenses.
1. Strong verbs. To change strong verbs to its '-ません' form, the 'う' sound is changed to an 'い' before 'ません' is added onto the end.
例えば： 書く = 書きません,　遊ぶ = 遊びません,  習う = 習いません,　買う = 買いません

2. Weak verbs. To change weak verbs to its '-ません' form, the 'る' sound is dropped and 'ません' is added.
例えば： 食べる = 食べません, 上げる = 上げません, 見る = 見ません, 教える = 教えません

3. Irregulars. Just have to know them!
例えば： する = しません, 来る = 来ません

-ない form (informal)
The 'ない' suffix is an informal way of making a verb negative.
1. Strong verbs. To change strong verbs to its '-ない' form, the the 'う' sound is changed to an 'あ' before 'ない' is added onto the end.
例えば： 書く = 書かない,　遊ぶ = 遊ばない,  習う = 習わない,　買う = 買わない

2. Weak verbs. To change a weak verb into their '-ない' form, the 'る' sound is dropped before 'ない'' is added.
例えば： 食べる = 食べない, 上げる = 上げない, 見る = 見ない, 教える = 教えない

3. Irregulars. They're... irregular.
例えば： する = しない, 来る = 来ない

How about past tense?

-ませんでした form (polite)
'ませんでした' is added on the end of verbs to denote a past negative.
1. Strong verbs. To change strong verbs to its '-ませんでした' form, the 'う' sound is changed to an 'い' before 'ませんでした' is added onto the end.
例えば： 書く = 書きませんでした,　遊ぶ = 遊びませんでした,  習う = 習いませんでした,　買う = 買いませんでした

2. Weak verbs. To change weak verbs to its '-ませんでした' form, the 'る' sound is dropped and 'ませんでした' is added.
例えば： 食べる = 食べませんでした, 上げる = 上げませんでした, 見る = 見ませんでした, 教える = 教えませんでした

3. Irregulars. I think I've said enough about irregulars and how they're irregular.
例えば： する = しませんでした, 来る = 来ませんでした

-なかった form (informal)
To express a negative in informal speech, '-なかった' is added on the end.
1. Strong verbs. To change strong verbs to its '-なかった' form, the the 'う' sound is changed to an 'あ' sound before 'なかった' is added onto the end.
例えば： 書く = 書かなかった,　遊ぶ = 遊ばなかった,  習う = 習わなかった,　買う = 買わなかった

2. Weak verbs. To change a weak verb into their 'なかった' form, the 'る' sound is dropped before 'なかった' is added.
例えば： 食べる = 食べなかった, 上げる = 上げなかった, 見る = 見なかった, 教える = 教えなかった

3. Irregulars. Irregulars will always be irregulars.
例えば： する = しなかった, 来る = 来なかった

Verb endings also change to reflect past tense. Let's have a look at past tense in both polite and informal ways.

-ました form (polite)
By changing a verb to its '-ました' form, it effectively changes the verb from present to past tense.
1. Strong verbs. To change strong verbs to its '-ました' form, the 'う' sound is changed to an 'い' before 'ました' is added onto the end.
例えば： 書く = 書きました,　遊ぶ = 遊びました,  習う = 習いました,　買う = 買いました

2. Weak verbs. To change weak verbs to its '-ました' form, the 'る' sound is dropped and 'ました' is added.
例えば： 食べる = 食べました, 上げる = 上げました, 見る = 見ました, 教える = 教えました

3. Irregulars. They're tricky little things!
例えば： する = しました, 来る = 来ました

-た/だ form (informal)
Changing a verb to informal past tense is a little trickier, as the endings can change with different verbs.
1. Strong verbs. To change strong verbs to its '-た/だ' form, the the 'う' sound is changed before 'た' or 'だ'' is added onto the end. The rules to strong verbs are a little obscure.
例えば： 書く = 書いた,　遊ぶ = 遊んだ,  習う = 習いだ,　買う = 買うんだ

2. Weak verbs. To change a weak verb into their '-た/だ' form, the 'る' sound is dropped before 'た' or 'だ'' is added.
例えば： 食べる = 食べた, 上げる = 上げた, 見る = 見た, 教える = 教えた

3. Irregulars. Damn these outcasts of the verb group! xD
例えば： する = した, 来る = 来た