Okay, everyone hates kanji. Even Chinese people hate kanji (because in Chinese, every character has more or less only ONE pronunciation). And the dictionary definitions for kanji are really bad, so you don't know why x is also used for y which seems like completely different meanings to you. In reality kanji are pretty easy.

First off, each kanji only has 1-2, very rarely 3, individual pronunciations and/or meanings worth knowing. In most texts, anything more than those 2-3 pronunciations — or any rare kanji, or kanji not taught in school, even if it's standard pronunciation or actual common words, — will almost always have hiragana (furigana) to the side showing the correct pronunciation. Even usually in newspapers, which are the hardest things to read in Japanese (aside from archaic stuff). The exception is often really famous words, ex. certain place-names, but even those actually might also be written in/with hiragana instead. Stuff written for highschoolers and younger (meaning: most manga) will ALWAYS have pronunciations on kanji, even if they're super common kanji. (About the list of kanji people learn in school: by adulthood Japanese people have forgotten hundreds of them, but they know hundreds more that are outside of the list.)

Second off, often a kanji really only actually has 1 meaning but what you're seeing is that it's short for a longer compound word, in which case you have to remember that too. Generally speaking, memorize 2 meanings and 2 pronunciations per kanji and you'll be completely fine. There's only 2,000-3,000 kanji in actual, real (not "look I picked a cool unknown word/kanji from the thesaurus") use; that's 6,000 meanings and 6,000 pronunciations. You can learn pronunciation by reading manga, watching anime with Japanese subtitles and even installing furigana-insertion add-ons; so just focus on meaning instead. After around 1,200-1,500 kanji (based on my own experiments) you can read most stuff comfortably.

Now you need to know that a kanji (and all Japanese words/phrases/sentences in general) can ALWAYS have a NON-LITERAL meaning, for example 山 "mountain" can also be used like "a mountain of chores". Secondly, two kanji together almost always mean "x and y at the same time":

 

子 child 犬 dog = 子犬 a child and a dog at the same time = a puppy (not "a child's/childish dog")
病 sick 弱 weak = 病弱 sick and weak at the same time (not "sickly weak")
重 heavy 大 big = heavy and big at the same time = (non-literally used to mean) important, serious
部 section 屋 indoor = a section (of a building) and indoors = a room (not "a sectioned room")
夫 husband 婦 wife = husband and wife = married couple
特 worthy-of-naming (= special) 別 apart, (non-literally: tell apart, distinguish) = special and apart (from others) at the same time = exceptional, specific, peculiar etc
紅 bold red 白 white, unmarked (ex. blank paper, purity) = red and white at the same time (not "a mix of the two, reddish white, pink")
回 turn, (travel) around 答 answer = turn and answer at the same time (= reply/comment back)

Here's an example sentence:

• 「池」は、地 面 に できた くぼみ に 水 の たまった ところ。ふつう、湖 沼 より 小さい もの。
　"Pond (池)" is a place/situation where a dip/hollow in the ground (地) surface/face (面) has been made by water (水) accumulation. Normally, compared to "lake" (面) "swamp" (沼) it's a small thing."

While "lake-swamp" (= lakes and swamps, not a lake that's also a swamp) is easy from context, we should think of "ground-surface/face" as "ground" and "surface" at the same time = the ground's surface".

The most frequent problem is that you'll see something like 時 "time", 代 "age, era", 時代 "time, age, era" in the dictionary: so you won't know the difference between 時 / 代 (simply time, simply era) and 時代 (time and era all at the same time like a package). The meaning is actually just the literal one.

When they don't clearly have "and" the first ones are details to the second ones. This type is a lot less frequent:

低 low 血 blood 圧 pressure ("low" is a detail to "blood pressure"; "blood" is a detail to what kind of "pressure" it is)
白 white 杖 walking stick = blind man's white cane
両 both 肩 shoulder(s)
絶 cut off 望 hope = (fall into) despair

It's hard to explain but the two are actually the same thing. When we put kanji together we're signalling that we see the ideas as a SET, inseperable from each other. We're not talking about a "white cane" (ex. temporarily painted white), which would be 白い 杖 ("is-white cane"). We mean the "white" and the "cane" are like a set package; no "is, was, will be" white but "always, unquestionably" white. We're not talking about a temporary low pressure, we mean the entire idea of having low blood pressure as a set idea (the ideas of "blood, low, pressure" being unseperable from each other). With "both shoulders" your SET of shoulders.

However that's difficult for most people so you can continue to think about it as being just plain description if you want.

4-kanji words can often be thought of as 2 words with "is" in-between:

責 responsibility 任 entrust 重 heavy 大 big = (the) responsibility entrusted (is) heavy (and) big = a grave responsibility

Examples of cases where you have to realize a kanji/word is actually missing (these are, relatively speaking to the "straight-meaning" ones, few — but still SEEM like a lot):

通 through 訳 translate = interpretation (audio translation) = must be something like 通 through  耳 ear 訳 translate.
少 few 女 female human = girl = must be 少 few 年 years 女 female human.
今 now, this 回 turn around = '"this (time) around" next time, last time = must be something like 今 this 時 time/when 回 (turn) around.
贅 luxury, 沢 marsh = "extravagance, luxury" = actually 贅 luxury 沢山 a lot/mountain of = "a mountain of luxury".

The problem with these isn't always the compound itself; some kanji can ALWAYS be short for a longer word, for example 学 "learn" can ALWAYS be short for 学校 "school-(house)", (中学校 —> 中学 middle-school), 大学校 "university" can be shortened to just 大 (東京 大学校 —> 東大 "Tokyo Uni"), and in the first place 中学校、大学校 are really short for something like 中教学校 "middle instruction/-ism learn-building" (just plain "middle-learn" doesn't make much sense).

If the kanji used makes absolutely no sense, it's been written that way for its SOUND and not its meaning. These are, comparatively speaking, really rare and since they're completely obvious (because, well, they're nonsense) when you find them, and their pronunciations are usually explained with hiragana/furigana, don't even think about them.

If there's two kanji with seemingly the same meaning, there are also always Japanese people who don't know the difference so you can easily find explanations online (and you don't have to worry about mixing them up).

Finally: never "memorize" a multi-kanji word unless you know for a fact it's "missing" a kanji or written for sound and not pronunciation. It's a complete waste of time and brainpower to memorize "puppy" when you already know it by sight and context (because it's actually "child dog"). Japanese people do not see something like "child dog" as a unique, individual word like our "puppy": they see it as two separate words they can mix and match whenever they like. That's how kanji are used: like LEGOs. So that's how you have to learn them. A good example is the word "keigo" you'll see when people write about politeness levels in Japanese, as if it's untranslateable: this is literally just 敬 respect 語 single word, language = "respect-showing word(s)".
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Here below, I'm going to put a few very clear examples (as I find them) of why you NEED to learn individual kanji and completely ignore whatever your translation dictionary may say about the resulting compound.

1. In the default Macintosh Japanese-Japanese dictionary (スーパー 大 辞林) we find the following:

はい【拝】 hai
① 頭 を 下げて 敬 礼 する こと。「～ を 行う」
To lower one's head and give "respect (and) thanks". [Note: It's very possible this is the origin of "hai - I'm listening to you, that's correct, yes..."]

② 手紙 文 で，自分 の 名 の 下 に 書いて，相手 に対 する 敬 意 を 表す 語。「中村 一郎 ～」
In letter-sentences, writing below one's own name (= at the end of the letter), a word that shows "respect meaning" towards the other person.

はい どく【拝 読】 hai doku
① 読む こと を へりくだ って いう 語。「お 手紙 ～ し ました」
A word meaning to "lower oneself in respect" and do the event of reading.

In the default Macintosh JP-EN-JP dictionary ( ウィズダム 英 和・和 英 辞典 ) however:

【拝】 sincerely (yours), pray (with a bow), worship, receive an imperial order, see*.
【拝 読】 read(ing)

Checking all the other JP-EN dictionaries I had access to they had exactly the same definitions, just reworded (Goo, Japanese Learner's Dictionary [dictionary.j-cat.org], Jisho.org [Edict], Weblio). Wiktionary was the same while also missing half the definitions for 拝. *One or two dictionaries noted that it's "see someone/something of high status", most didn't.

The actual sentence I saw this word/kanji in was this:
ブログ 拝 読 し ました　"blog 拝 読 did" = (I) read (your) blog. Matching formality level: I had the pleasure of reading your blog.