What is the best way to learn and memorize kanji?

The best way to learn kanji is to learn them through common words rather than one by one, notice their components (many kanji share parts), learn the basic stroke order rules, and review with spaced repetition. Start after hiragana and katakana, with the roughly 447 kanji used in JLPT N5 words.

How many kanji are there?

Japan’s official list of kanji for general use, the jōyō kanji, contains 2,136 characters (revised in 2010). Newspapers and everyday texts rely mostly on these. You do not need all of them to start reading – the kanji in JLPT N5 and N4 vocabulary already cover many everyday words.

A beginner’s method

Kana first. Learn hiragana and katakana; you need them for readings and word endings.
Kanji through words. Learn 学生 (gakusei, student) and you have 学 and 生, plus a reading of each.
Use components. 休 (rest) is 亻 (person) + 木 (tree) – a person resting against a tree. Every kanji page lists its radical and components.
Know the stroke order rules. Top to bottom, left to right, horizontal before vertical. The stroke order lookup shows the exact order.
Recognise before you write. Reading is what you need most; add handwriting for the kanji you use often.
Review on a schedule and test yourself with a kanji quiz.

Why do kanji have several readings?

Most kanji have an on’yomi (a reading borrowed from Chinese, used mostly in compounds: 学生 = gakusei) and one or more kun’yomi (native Japanese readings, often with kana endings: 学ぶ = manabu). Learning words, not lone kanji, tells you which reading to use.

Sources

Agency for Cultural Affairs, Japan – Jōyō Kanji Table (常用漢字表), 2010 revision.
Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132(3), 354–380.