A couple of other obvious points:
Does your cube turn easily? Some cubes are just horrible, and even if you lubricate them, they're still difficult. If you haven't got one, a dedicated 'speedcube' is well worth buying.
If you're using a beginners method, it may be worth looking at other beginners methods, some are potentially quite quick, despite their ease of learning, whereas others take far too many 'baby steps'.
And you don't actually need to use 'finger-tricks' (single-finger flicking turns, rather than holding and turning) to get below 1 minute, even with beginners methods. Just practice, as stated before.