AH, kanji! I'll break this down as if you're a total beginner:
Kanji are extremely useful and if you're going to do ANYTHING with Japanese, you need to know them.
Each character will usually have two ways to read them: a "Chinese" reading and a Japanese reading. Some may have many more, some have far fewer.
Most of the time, if you encounter two or more kanji together, it will be "Chinese Reading+Chinese Reading." Exceptions are usually Japanese surnames (mostly JR+JR) and some nouns and verbs.
As for actually studying them, your ultimate goals may dictate what you want to do. If you're traveling in Japan and focusing on day-to-day conversation, you may want to only worry about recognizing certain characters. If you'll be expected to read and write Japanese fluently because you're living in Japan or working in a field where you will be using them a lot, you might want to go over them in depth.
Suggested method:
Buy a graphing notebook. Write a large entry character (LC square shows a 3x3 space being dominated by one character), its meaning, and its readings. Write the character stroke by stroke, each repetition adding a stroke until you have completed the character. Then write the character 20-30 times. Beneath all of that, you can write some vocabulary and practice sentences:
LCLCLC C Reading stroke1 stroke 2........
LCLCLC J Reading kanji1 kanji 2 kanji 3......
LCLCLC meanings vocab1 vocab 2 vocab 3
Sentence 1. Sentence 2. Sentence 3.
Use a flashcard app that has an spaced-repetition system and review them often. Be sure to review uncommon readings as well! I stress flashcard apps because if you have your smartphone or tablet with you, you can sneak in some practice while you go about your life; waiting in line, during your lunch break, while using the restroom...
Next, practice reading them! Your textbook is great for training wheels, but try reading untranslated manga, which are usually fairly simple and engaging. I suggest Yotsuba& for starters. Some series like One Piece and Naruto should be fairly simple since they're aimed at 12-year-old kids (available on sites like Amazon). NHK WebEasy is another decent resource. Try reading Japanese news sites, books (proper novels) and magazines that are aimed at adults once you've advanced far enough.
Try writing in Japanese, both by hand and using a keyboard. If you're taking Japanese in school, you'll have a load of this. If you're going it alone there are pen pal sites/apps that will connect you with native speakers who will correct your mistakes (feedback may be varied - the average person may not have a sophisticated grip on WHY something is incorrect and provide less than stellar feedback).
PACE YOURSELF - kanji are a marathon, not a sprint. There are about 2,200 kanji used in Japanese publications. You probably won't master them all at once unless you dedicate an INSANE amount of time to the task. There's a few ways you can choose where to start, but I suggest getting a hold of which characters are used in the JLPT (Japanese Language Proficiency Test). The N5 test (lowest level) tests about 100 simple, basic, common characters. Each level adds more characters, in increasing complexity and rarity, until you hit the 2,200 mark.
Another route is to start with the kyoiku kanji - the 1,000 or so characters learned by Japanese schoolchildren until 6th grade. Again, they increase in number and complexity as the years go on, but mastery will guarantee about 90% literacy.
In any case, I'd aim for studying 15 kanji a month to start and increase it to about 20 when you're good on about 300 of them. As you learn more they DO get easier - some characters are used MOSTLY in one way (that is, you'll usually see them either stand alone or, on the other hand, mixed with other kanji), or only with one other character.
Lastly, learn about radicals and the basic components of kanji. There's usually a semantic component (gives a clue on what the thing is related to) and a phonetic component (how to read it the Chinese way). This trick can help you maybe 85% of the time. This link provides a lot more information about the background of kanji: http://www.kanji.org/kanji/japanese/writing/outline.htm. Some of it is admittedly superfluous for the average person's needs