Springs in the "skeleton" are vital to quality cubing. You can sense if there is one by pulling on a layer. It should have a "springy" give of 1 millimeter or so. New cubes can be tight and hard to pull apart, but it's still a different feel than a springless cube. Do NOT buy a springless cube! Modern cubes are mostly built so you can't get to the springs and adjust them, which is a shame. The exception are the DIY cubes you can order from rubiks.com and other places. Not that it's a must have at all, but you can get slightly better results by adjusting the spring tensions, and even replacing broken springs.
Though some cubes can get pretty good without treatment, I recommend that you lube it. Many lubricants work. I used candle wax first. It works fine, except that it needs to be reapplied twice a week. Later I turned to silicone spray, which works for months. It is what all speed cubers use these days. You can get it cheaply at any automotive store.
One weird thing with the silicone spray is that when you first apply it, on some cubes, it will act almost like glue, and will be very hard to turn for about a minute or two, and only after that will it be easy to turn.
In the 1981 Swedish championships, the cubes were silicone treated, but not "broken in", so we only got about half out normal speed. I won with 40.43 sec...
The cubes I can find now (made by Oddzon (buy here)) are good and fast mechanically, but have horrid paper stickers (under a thin plastic membrane) that wear out in no time. Some people paint them when the stickers are gone. I use regular colored vinyl film. It works fine, once you've found 6 colors of film. And you get enough vinyl for a 1000 cubes :-) These days you can even order high quality replacement stickers from cubesmith.com
Even the best cube will wear out. That is natural, just buy a new one. I have used up over 15 cubes over the years.
I'm right handed, but I use the left hand for turning, and the right hand for more or less just holding the cube. I don't know why. It took me several months before I realized I was doing this "left handed".
Use a firm but not too tight grip.
Every sequence of two (often more) moves can be done with column moves,
that halves the number of movements you need to make. It takes a while to
get to see sequences as moving columns in stead of separate turns, but it's
very easy once you "get" it.
When it is about 3/4 complete, you can turn the top layer with the index finger. Do the second turn that way. Meanwhile, start gently turning back with the rest of the left hand, still holding the grip between red and yellow. When the second turn is 3/4 done, you will be able to do the third turn instantly.
This can be done ridiculously fast, 0.1 seconds or so. But start practicing slower...
You need a cube in good shape, or you will break it or get stuck.
This is Sune™ at top speed. Seven moves in 0.70 seconds. It's in 30 frames per second. In frame 9 more than one full turn is done! To see anything of what's going on you need to step through it. If that's too confusing, first look at the slow version.
This is the same Sune™ done slower to illustrate the finger tricks. The first three moves is a trigger, just as described above. The next two moves is also a trigger, but skipping the first move. And the final two moves are a simple column sequence.