I don't have much experience with Rubik's Cubes, but this is a good place to start. The web page is literally titled "A Rubik's cube solution
that is easy to memorize":

http://beust.com/rubik/

But many people learn better with video, so here is a video tutorial, entitled "Easiest 3x3 Rubik's Cube Tutorial You Will Find":

http://www.metacafe.com/watch/818522/
Source(s):
http://beust.com/rubik/
http://www.metacafe.com/watch/818522/

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Try looking at the Nintendo DS, they have a game called 'Rubik's World'. And it has tutorials and games for you to play to help you solve a Rubik's cube.

It may be the best option in terms of a traveling companion, as it doesn't require a connection to the Internet.
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Well i found these sites to be very helpful in making you get up t speed in solving Rubiks Cube. Peter still hq dot com is essentially a beginners walkthru on solving the cube and it has some good links plus a pdf explaining the process.
There is the second site by Lars Petrus the 1981 swedish champion who finished 4th in the 1982 world Championships his method essentially is solving for speed and his website is well broken down into a)Intro b)Basic idea c)method which consists of a seven step point by point guide on how to do it .The site also has a java based game where you can harness your skills.
You can also check out the following videos from metacafe dot com for solving the 3 by 3 and 4 by 4 Rubiks cube. I hope this information is helpful to you and good luck.
Source(s):
http://peter.stillhq.com/jasmine/rubikscubesolution.html
http://lar5.com/cube/
http://www.metacafe.com/watch/820051/the_easiest_4x4_rubiks_cube_tutorial/
http://www.metacafe.com/watch/818522/easiest_3x3_rubiks_cube_tutorial_you_w...

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This is a tutorial I've used. Hope it helps.
Source(s):
http://www.howtodothings.com/hobbies/how-to-solve-a-rubiks-cube

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I personally can vouch for Lars Petrus' site:

http://www.lar5.com/cube/index.html

There is a great turorial that I used to learn to solve the rubiks cube while I was in college. It breaks it down into 7 sub steps that you can learn individually. Once you've mastered them you will be able to start working on some shortcuts or on combining steps to solve it even faster.

There are a lot of interactive tools on the site to teach you each step as well as discussion on some of the rubiks cube culture.
Source(s):
http://www.lar5.com/cube/index.html

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Primarily, the cube can be solved in a layer by layer approach using only 4 algorithms. An algorithm is a sequence of moves (a turn of one of the six sides).

When you have memorized the four algorithms and the situations in which to perform them, you will be able to solve a rubiks cube within one minute. This is not a joke! (you do need to practice a little and of course you will need a reasonable cube for this).

Learning the algorithms is the hardest part.


RECOMMENDED WEBSITE---------------

http://vanderblonk.com/cube/yy/

HERE ARE SOME OF MY RECOMMENDED TUTORIALS---------

ITS A 5 PART SERIES---------
HERE ARE THE FIRST 3-------

http://in.youtube.com/watch?v=LcJ-y39GzzQ

http://in.youtube.com/watch?v=Xfw045MjVDQ

http://in.youtube.com/watch?v=RzvNseyBhyQ
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