A good stickerless cube is $10.99, on Amazon. A 4 by 4 is also sometimes called the Rubik's Revenge. A magnetic 4 by 4 on Amazon is $14.99. You would want a magnetic cube because it is not loose, moves very quick, and has a great feel. You might not think that that is necessary to pay $14.99 when you can pay $10.99, but if you are really into cubes and getting fast this is probably for you.
There are four different types of pieces on the four by four. The corner pieces are located in the corner. Four per side and each one has 3 colors. The second group of pieces are the center pieces. There are four of them in the center of each face. The last remaining pieces are the edge pieces which are on the edge of two faces. 8 edge pieces per side.
When solving a 4 by 4, you are basically converting it into a 3 by 3. The first step is to make the centers. (the center 4 pieces on each side) The second step is to group the edges. The last step is to solve it like a 3 by 3. There are 2 cases that can't happen on a 3 by 3 that sometimes do on a 4 by 4, so you'll have to learn those. They are parities. (see parity section below)
How to solve with a video: HERE
This is the OLL parity. You solve both OLL parities the same. You recognize it because when solving the yellow cross you have a non-3 by 3 case. (not line, angle, or dot) It is when you have 1 or 3 incorrect edge pieces.
To solve it, hold 1 of the incorrect edges at you, then do this algorithm: (All R, R', and L moves are with 2 layers) R, U2 make the face on the bottom at you, R U2, R, U2, R', U2, L, U2, R', U2, R, U2, R', U2, R'
This is the OLL parity. You solve both OLL parities the same.You recognize it because when solving the yellow cross you have a non-3 by 3 case. (not line, angle, or dot) It is when you have 1 or 3 incorrect edge pieces.
To solve it, hold the incorrect edges at you, then do this algorithm: (All R, R', and L moves are with 2 layers) R, U2 make the face on the bottom at you, R U2, R, U2, R', U2, L, U2, R', U2, R, U2, R', U2, R'
You solve both PLL parities the same. You can recognize this by when your cube won't solve. It can look like in image 1, where there is 2 incorrect edges next to each other. It can also look like in image 2, where there is 2 incorrect edges across from each other.
To solve it, hold one of the not solved sides at you, yellow up, (example: in the picture above, hold the side with the red center and 2 green pieces at you), and do this algorithm: 2R2, U2, 2R2, u2, 2R2 u2.
You solve both PLL parities the same. You can recognize this by when your cube won't solve. It can look like in image 1, where there is 2 incorrect edges next to each other. It can also look like in image 2, where there is 2 incorrect edges across from each other.
To solve it, hold one of the not solved sides at you, yellow up, (example: in the picture above, hold the side with the red center and 2 orange pieces at you), and do this algorithm: 2R2, U2, 2R2, u2, 2R2 u2.
The 4 by 4 super cube is like the 3 by 3 super cube except harder. It is a super cube, because each side has arrows going in one direction. This makes it harder, because not only could a 4 pieced center be twist, but just one of the little center pieces could be twisted.
If you want to get faster at the 4 by 4, you should try to get fast at the 3 by 3 first. You should do that, because when you are in 3 by 3 mode, you want to be fast at that. The fastest method for the 4 by 4 is called the Yau Method, and it is really easy to learn. After you have learned the Yau Method and you still want to improve, you can learn thins like 3-2-3 edge pairing, half centers, OLL and PLL parity tricks, and more. To learn the Yau Method, go to the Yau Method sub-page on this website.