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?
CK2: technology screen
crusader-kings-2
Related Solutions
First and Foremost, you should stop playing as a count, it is no longer as straightforward as it used to be. By playing as a weak starting character, you are intentionally handicapping yourself for a game you were completely unprepared for. Try playing as a french duke to begin with. That means that you will have the power of your liege for defensive wars, and can declare war on people much weaker than you, ie rebellious counts and weaker dukes. You should start out only going to war with people who you outnumber by 50 percent, minimum, since you have to deal with seiging and defended bonuses.
You should try not to rely on warfare. Making the right marriage matches and waiting up to an hour for them to come to fruition is part of the game, (playing the long game is part of all paradox games really). In fact, Marriage is actually a much stronger mechanic for taking territory than combat. Not only that, but marriage is the most powerful mechanic in combat as well, because it allows you to call allies with vastly larger militaries than you can command early in the game. An alliance with the king of NearbyNation will let you call in thousands or tens of thousands of additional troops.
Still, the overall point is to stop playing as weak characters. Counts and the occasional 1 province duke in ireland are not advisable start locations for someone unfamiliar with paradox games, because like real life, paradox combat is very swingy. A 20 percent difference in starting numbers can be a 200 percent difference in casualties. That means when starting out, make sure you absolutely have a substantial numerical advantage against whoever you declare war on.
As for your example of losing a battle where you had superior numbers, you probably made a mistake with regards to combat morale, but its difficult to know without screenshots. Even if they had a vastly better general you should not lose a 1000 to 2500 fight, unless you had virtually no morale (probably from not paying the men).
Basic Concepts to Practice:
Revoking a county title as a Duke. This usually starts a war of rebellion that you can crush.
Invading an independent county as a Duke. This includes ones rebelling from their liege that have become independent temporarily
Marrying daughters into a powerful alliance (nearby, but not where you want to expand to)
Marrying sons to inherit land
Keeping your most important vassals and advisors (particularly the spymaster) happy.
The CK2 Wiki has an excellent and up-to-date article explaining technology. It has this to say about technology spread:
Technology spreads between provinces that border each other, as well as between provinces in the demense of the same ruler even if they do not border each other. The rate of growth depends on the degree of the technology lead in the more advanced province. The Spymaster's Study Technology mission also lets technology spread from the target province. The other advisors' research missions also increase the speed of spreading, as do the School and University buildings.
So by sending your Court Chaplain (or Steward, or Marshal) to research tech in your capital, you increase the rate at which the capital increases in cultural (or economic, or military) technology and therefore increase its lead over provinces neighbouring your capital, which in turn increases the rate of spread of tech from your capital to its neighbouring provinces and non-neighbouring provinces in your personal demesne.
This is a good thing! Your capital is where tech will increase the fastest, so you want to encourage those advances to spread to other provinces in your realm. If your capital borders directly on provinces outside your realm (for example, Krakow in Poland borders Rus provinces and Hungarian provinces), either accept that you're helping your rivals somewhat, or move your capital to a more central province.
If you want a particular province to catch up faster instead, sending your councillors there will multiply the effect of spread from its neighbours.
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Best Answer
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