How to get faster: an introduction to speedcubing
Solving the cube in under a minute with the beginner solution? Ready for more?
First, note that learning algorithms alone will not make you faster. In order to get faster, in the words of a great cuber, SLOW DOWN and LOOK AHEAD. Fast fingers and a huge algorithm bank are worthless if you spend half your solve time looking for the pieces you need. Slow down your moves and use the time to look and plan ahead for your next moves. It seems counterintuitive, but over time this will help you get faster. This will help you at any level, whether you're a beginner looking to break a minute or a sub-20 cuber trying to shave off a few seconds. Practice, practice, practice!
The beginner layer by layer tutorial is a good first step to the Fridrich method. If you want to continue with the Fridrich method, start by learning the first seven algorithms on the OLL page, as well as the 21 PLL algorithms which will allow you to permute all the last layer pieces in one step.
You'll also want to start working on an advanced F2L, wherein the first layer corner and adjacent middle layer edge are solved together rather than in separate steps. I have provided algorithms, but I recommend playing around and trying to figure things out on your own rather than blindly memorizing them. Most of the F2L algorithms can be derived intuitively as long as you understand what's going on.
Once you've reached around 20 seconds on average, you can memorize the remaining 50 OLL algorithms. It should be noted that sub-20 is well within the realm of possibility without knowing them all though.
Of course, Fridrich isn't the only speedsolving method out there. It's just the one I'm most familiar with. Popular alternatives to Fridrich include the Petrus and Roux methods. These use different approaches to solving the cube but are worth looking into if you're interested.