Tips
I realise there are already a lot of websites that give tips for new cubers, so I will try not to repeat the same list here. Instead, I will just try to give you the most important ones; the tips that helped me improve my times the most. Some of these tips are aimed at speedcubers that are learning the Fridrich system, but in a lot of cases, they will also be useful for cubers with other methods.
1. Solve the cube very slowly from time to time
What this means is, you should solve the cube very slowly sometimes, without timing yourself. When I say 'slowly' I mean very slow, actively thinking about every move you are making. When you are solving a 'sub-problem' (for example: a corner-edge pair), try to find a way to minimise the number of moves and cube rotations needed to solve that problem. The number of moves is not the most important factor to decide whether a solution is good or bad, but I recommend newbies to actively spend time on minimising the number of moves. There are many shortcuts for solving corner-edge pairs, when the F2L is not completely solved yet. Try to find these on your own while solving the cube slowly.
2. Slow down during practice!
Speedcubing videos might give you the impression that speedcubing is all about speed. But that's not true. In my opinion, speedcubing is all about finding a balance between fluency and speed. You should not focus on speed too much. It's more important to work on your technique. The speed will come naturally. When I am having a bad cubing day, one of the things that helps me to get better times is telling myself to slow down. The idea is not to solve the cube at full speed (especially the F2L), but to go a little slower instead. This will enable you to watch the pieces, and it gives you time to look ahead. When you are solving a 'sub-problem', you can try to figure out what your next step is going to be. For people that use Fridrich, this means that when you are inserting a corner edge pair, you should have already spotted the next pair. Also, after finishing the cross, you should at least have a general idea about where your first corner edge pair is. The goal of looking ahead this way is that you can solve the cube without, or with only very small delays between algorithms.
If you don't believe me, just time yourself, while aiming at your normal average plus 5 or 10 seconds (this will probably work for most people who average around 17-30 seconds). You will see it's not that hard to do that. Do about 10 of those very slow solves, and try to look ahead while doing this. Now, slowly accelerate to your normal times. I hope you'll find that rushing through a solve like a mad man it completely unnecessary, and that you can have the same (or even better) results by slowing down just a little bit.
One last thing: It's a good idea to solve the last layer at full speeds, because when solving the last layer, it's almost impossible to look ahead. It's also a good idea to practice solving all different steps at full speed, because you should be able to do them without thinking. Of course, I am not telling you to go slow, just go as fast as you can without making mistakes, and without big delays between algorithms.
3. Tips on memorizing algorithms
If you are just starting with cubing, the number of algorithms you have to memorize can look overwhelming. Indeed, if you are trying to learn the whole Fridrich method, you do have some memorisation to do. Fortunately, memorizing a sequence of speedcubing moves cannot be compared with memorizing a sequence of digits. It is actually much easier. Here are some tips that can make memorisation of algorithms easier.
For F2L algorithms: know where the pieces are going
Most F2L algorithms are very easy to understand intuitively. Most of them start with 3 or 4 moves to pair up the corner and the edge, and end with 3 or 4 moves to insert that pair in the correct slot. Basically, if you know the moves to pair up the two pieces, the rest of the algorithm is trivial. So don't try to memorize them as sequences of individual moves. Just look where the pieces are going. This means: look where the pieces are going when you are memorising the algorithms. During speedcubing, the ultimate goal is not to look at the pieces you are solving, but to look ahead to see the next corner-edge pair. Another advantage of knowing how the F2L algorithms work, is that when you understand the Fridrich F2L system, it's easier to find shortcuts.
For OLL algorithms: break up and fix the F2L
A lot of OLL algorithms are easier to memorize when you think of them as a way to break up the first two layers, and then solve them again. If you remember where the pieces are going (especially the F2L pieces), or where groups of pieces are going, memorisation of these algorithms can be easier.
In general: an algorithm is a sequence of 'elementary moves'
A lot of algorithms are easier to memorize when you think of them as a sequence of 'elementary moves', rather than individual moves. Some combos of moves occur quite frequently in algorithms, such as (RUR'), (RU'R'U), (R'FRF'). If you group these moves, and think of them as 'elementary moves', it makes memorisation easier. So instead of R2U'R'U'RURURU'R, you can write (R2 U') (R' U') (R U R U) (R U' R). Also try to remember what your hands need to do during each 'elementary move'.
4. The psychological factor
Well.. What can I say about this? I can't claim to be an expert when it comes to this subject, but I can give you some advice. You should never underestimate this factor. This factor has a pretty big influence, not only when you want to compete in an official competition, but also when you are just practicing at home.
Relax while cubing
Being relaxed and focussed during cubing is very important. If you make yourself nervous, or if the cube makes you mad by locking up all the time, just take a break (op #6 might apply to your cube). Cursing and swearing won't help you to get better times. Yes I know, I am also guilty, but I try to minimize this as much as possible. Listening to your favourite music during cubing also helps. Also, sometimes you just have to be satisfied with 'average' and 'mediocre' times. Even happens to the best cubers.
Practice with people watching
Practicing speedcubing with an audience is pretty useful. In these situations, it's always different than when you are just cubing all by yourself. Just be careful not to bother any people who are not interested too much.
5. Choose your algorithms wisely
A lot of cases have different algorithms associated with them. Look on different websites to find all the different algorithms, and then decide which one you want to use for speedcubing. When comparing two different algorithms, the number of moves is important, but not decisive. You should also consider the next things: Are there any cube rotations in the algorithm? Does the algorithm use any nice two-generator (R, U) moves? Are the moves easy to execute when you are under pressure (shaking hands)? Are the PLL algorithms stackmat-timer friendly? Is visibility on the cube during F2L algorithms good enough?
6. Use a good cube
Last but not least, you should use a good speedcube. Some people prefer cubes that are a bit stiff, some like cubes that are very loose. At the moment, I use a rubiks.com DIY cube. I prefer to adjust the screws so the cube is a little loose, and I regularly lube it with silicone spray. Experiment with this a little to see which cube you like the most.