This is a fast method to solve the 5 by 5 Rubik's cube, and it is the beginner's method (It was the first method I learned). This method has 3 steps: solve the centers, solve the edges and then solve the cube like a 3 by 3. You can look through the steps, and if it does not make sense, you can watch a video on each step. I hope you start solving your 5 by 5 Rubik's cube!
The centers are the inner 3 by 3 square in the middle of every side (picture shown of them solved on the left/below). You solve them 1 at a time in 2 parts. The first part is solving the cross of the 3 by 3 square, and the 2nd part is inserting the corners on every center. For a video on how to solve the centers press HERE
The first part in solving each center is to solve the cross of a center (shown above). To do this, you start by finding the first cross piece, then you rotate the cross piece up, so that it hits a solved cross piece, then, you rotate it over so that the rotation you just used now has no solved edge piece in it, then you used the same layer you already used and put that layer down (sometimes if the cross piece you are inserting is on the opposite side of the center you are solving you need to put a layer up and down twice). If you do not have a cross piece already in the center you are making then you pretend that there is one in any spot. P.S. You do this step on a center then the next step and then you move onto the next center.
Now that you finished the first step of your center you need to get the corners in to solve the center. you do this by putting a corner piece of the center on top of where it needs to go for example in the picture above it shows the solved white cross and the white corner is going to be inserted into the center by doing the algorithm. Then you perform this algorithm. (In this algorithm you do the R and R's with two layers), R, U, R', U, R, U2, R' to get a corner in it's spot. There is 1 different thing that can happen and that is when there is a white corner on the opposite side of your white cross, if this happens then you need to put the corner down (not onto a solved face) then rotate the corner piece over so that the corner is not in that same rotation and then put the same rotation you used before back up. If all of this did not help you you can watch this video: HERE
The edge is the group of 3 pieces between the blue and orange center.
In this step you will be solving 12 groups of 3 pieces (shown in the picture) called a edge. You solve each edge in 2 quick steps (like a center) and then you move onto your next edge. Once you solve all of your edges, you move on to the last step. There is 1 parity on solving the edges that is different from the way you solve the other edges and it mainly comes up on your last edge where the the most middle edge piece is twisted (look below for parity). P.S. I am not going to explain this step on the website because it is pretty hard to say in writing. If you want to learn, press: HERE
The parity on the 5 by 5 is while you are pairing the edges. In the picture shown, the parity is with the green-orange edge pieces. You can solve this while you are still solving the cube, or at the end. You hold the cube like shown and do this algorithm: (same as 7 by 7 parity algorithm and all R, R', L, and L' moves are with the 2nd layer only from their respective sides) R', U2, L, F2, L', F2, R2, U2, R, U2, R', U2, F2, R2.
P.S. Parity doesn't always have the rest of the cube solved.
You finish your cube by solving the cube like a 3 by 3. The centers that you made at the beginning are like the 1 piece in the center on a 3 by 3. The edges that you just solved are the edges that make up the white cross on a 3 by 3, and the corners with 3 colors on the 5 by 5 are just a corner on a 3 by 3. So now you can go ahead and solve your cube like a 3 by 3. There are no parities on 3 by 3 mode, unlike the 4 by 4.
If you want to get faster at the 5 by 5 Rubik's cube, you can learn the method Yau 5. This is the fastest method on 5 by 5. You also can work on your 3 by 3 step, and try to use a fast method like CFOP. To lean Yau 5, go HERE.