Notes on how to solve a Rubiks Cube. I like the Thoughtful Way better.
Terminology
R: the righthand face
U: the top of the cube
F: the side facing you
L: the lefthand face
D: the bottom of the cube
B: the back face, rarely used
i: inverse. Instead of the regular righthand rotation, rotate the face to the left.
x: You may need to repeat the sequence a few times to get the desired effect.
Algorithmic Way
This is from the instructions that were included with the cube with some annotation added by me. It's plugandchug but not easy to memorize.
Bottom Face

Pick a face. First solve the edges on that face.
 To flip the UF edge: Ri U Fi Ui

Now solve the corners.
 To rotate the UFR corner: Ri Di R D x1/3/5
Middle "face"
Now flip the cube so the solved face is on the bottom. Find an unsolved edge on the top that belongs in the middle. Rotate the top so that the edge just needs to be pulled down. If an edge is in the right place but oriented wrong, pull down another edge to eject it to the top face, then pull it down properly.
 Pull down front edge: U R Ui Ri Ui Fi U F
 Pull down right edge: Ui Fi U F U R Ui Ri
Top Edges Top
You want a cross on the top face. Ignore what's on the other sides for now; just form a cross on U.

If you see an L (two adjacent edges are correct), put the correct ones on F and R and do this to get a line.
 Solve Upper Cross: B U L Ui Li Bi (B means back, TODO: reorient this to get rid of B).

If you see a line (two opposite edges are correct), put the correct ones on L and R (so the line is horizontal, not toward you).
 Solve Upper Cross: F R U Ri Ui Fi

If nothing is correct, just apply the above moves until you get two adjacent or opposite edges positioned.
 If someone reassembled your cube incorrectly (1 or 3 edges are correct), this is usually the step where you discover that the only way to solve the cube is by disassembling it again.
Top Edges Sides

Now rotate U so that as many edges as possible are correct. Place two correct edges at B and R, then:
 Arrange Cross: R U Ri U R U U Ri
Top Corners
Now everything is solved except for the four corners on U.

If the some corners are out of place, put a corner that's in the correct spot in the upperfrontright position.
 Arrange Top Corners: U R Ui Li U Ri Ui L

Finally if all corners are in the right spot but rotated wrong, put a bad corner upperfrontright and apply the following sequence until only it is oriented correctly. Ignore the rest of the cube! It will look like everything is getting messed up. Don't worry; if you do this right, it will sort itself out by the end. When the corner you're working on is oriented correctly, turn U until the next bad corner is in UFR and do the sequence again.
 Rotate Top Corners: Ri Di R D x2/4
And you're done!
Thoughtful Way
This way is much easier to memorize because you figure out a lot more steps on your own. it's just method 2 on http://jeays.net/rubiks.htm You'll have to go through those instructions a few times before the following will make any sense.
Start
Do one entire face except for one corner. Leave that corner unsolved because you're going to use it as a scratch area.
Solve Mid Edges
You've got some edge pieces on U that you want to drop into place on the mid face. Choose one, but not the edge above the vacant spot  you'll leave that unsolved. Turn D to place scratch corner to be below the spot where the edge should go. Say you want to place the FR edge, turn D to put the scratch at FRD. Now turn U to where the new edge would take only a F or a Ri to drop it into place and:
 If it takes an F to place it, apply Ui Fi U F
 If it takes an R to place it, apply U R Ui Ri
You'll see that there's need to memorize those sequences; it will be immediately obvious how it works once you've done it once or twice. You just move the scratch corner up, slide the edge in, and drop it into place. Easy.
Repeat for 3 middle edges; don't bother solving the edge above the vacant spot. Now you have the D and middle face solved except for the scratch corner and the scratch edge right above it. TODO: it seems to make the final steps easier if you DO solve all 4 edges now? That could just be coincidence though.
Solve Upper Edges
You've got a scratch edge in the mid. Use it to buffer an edge down from U, spin U, and then move the edge back up into the correct spot. Solve all edges except for BR and UR (BDR is the scratch corner, BR is the scratch edge).
Now there's a good chance that all your edges are solved. If not, you'll have to memorize these moves:
 BR and UR are in the correct locations, but incorrectly oriented: B Ui Bi U Ri U R Ui.
 BR and UR are correctly oriented (the R color is correct) but swapped: Ui Ri Ui R Ui Ri Ui R Ui.
 BR and UR are in the wrong locations and incorrectly oriented: B U Bi U B U Bi U2.
TODO: those 3 sequences are by far the hardest and most obscure part of this solution. Is there any way to simplify this?
 It seems to be pretty easy just by fiddling to get any combination of edges into either the second state or solved. If swapped but oriented correctly, just hit Ui Ri Ui R twice. That's easy enough to remember.
Position Corners
Turn the cube so the scratch corner in DRB. Rotate U so the place where the scratch corner belongs is in UFL (the opposite corner). L D2 Li to swap the UFL with the scratch. Now the cube is pretty messed up, don't worry. Rotate U so the place you want to put the new scratch corner (the corner currently occupying DRB) is in UFL. L D2 Li to swap again. Keep swapping until all the corners have been moved to the right spot.
 To swap UFL and DRB: L D2 Li
What if the scratch corner, by coincidence, is correct? Swap it anyway. The scratch corner is never correct, even when it looks like it is!
Rotate Corners
Corners must be rotated in pairs, one clockwise and one counterclockwise. Choose two corners. If the corners are on opposite ends of the cube (don't share a similar face), turn R2 to put them on the same face, rotate them, then R2 again to fix things when you're done.
Put the first corner you want to rotate in UFL, then:
 To rotate it clockwise: L D2 Li Fi D2 F.
 To rotate it counterclockwise: Fi D2 F L D2 Li. (just the opposite turns played backward)
Now the cube looks all messed up. No worries. Just turn U so the other corner you want to turn is in UFL and then apply the opposite sequence. For instance, if I wanted to rotate UFL clockwise and UFR counterclockwise, I would apply the clockwise sequence, then Ui to move UFR to UFL, then the counterclockwise sequence, and finally U to put things back.
To memorize: notice that you only ever touch the L and F faces. You start with Lclockwise and Fcounterclockwise, so it's easy to remember how to start. And the pattern is easy: X D2 Xi Yi D2 Y, where X is the face you start with and Y is the other face.