The tips I've seen suggest that I should be able to focus my attention on particular areas within the technology screen, but I don't see how to do this. I click on various things, but nothing seems to happen unless I click on the thing that speeds that area to completion (costing me gold I think). Is that all I can do there, or is there some additional way of "focusing" attention?
You may have read tips relating to an earlier version of the game.
Now, if I recall correctly, technology advancements were more or less random, although you could set a focus area for each group (e.g. "light infantry"). This has been changed, and you now have more direct control over the technology.
Similar to Europa Universalis, you gain technology points over time in each of the three groups. The amount of points gained, depends on your ruler, your council, ahead/behind bonuses etc.
When you have enough points to spend, you click on the "advance technology" button – it's the button with the magnifying glass and arrow in the image below. Note, that technology doesn't cost you gold – just arbitrary technology points, which have no other use in the game.
Now, just because you have points to spend, doesn't mean you should. Since every level increases in cost, it would often make sense to save up for a more expensive advancement.
Also, some technologies will slowly improve on its own. For example, you might get a passive increase to "Church Infrastructure" by building a bunch of religious buildings. You can see that by hovering over the cogs, which should be spinning and have a slightly different colour (see screen shot).
Image borrowed from http://www.ckiiwiki.com/Technology