I've done area procedural and radar (long ago) and am currently on tower.
There is no quick way to make your strip scan effective, it's a bit like learning a new language. The more you do it the more it will become second nature.
There were different strip scanning/usage modes depending on whether you are on the procedural or planning position, or the radar position. I'm referring mainly to the planning position.
In the initial stages look to the most important items on the strips, which in the case of the units I've generally worked at was the level bay, and the time. This is formative in spotting conflictions. You are using the strips to form a mental picture of what's happening. After a time you should be able to reconcile the info on the strips to a surprisingly accurate picture of what is on the radar, right down to a/c passing at the time you've calculated, plus or minus a few seconds.
The best method I found was to follow the "strip trail" (which could comprise 3 or 5 or more strips, depending on the a/c and the route and the size of your sector) for each individual flight over its route, and then (constantly keeping an eye on the clock - this is a 4D exercise) for all the other flights, in turn. The priority is on the flights that will be conflicting first, then those later in time.
My job as a procedural or planning controller was to alert the radar controller to potential conflictions (ie: anything not going to be procedurally separated on or shortly after entering the sector) and regulate, by placing level/route restrictions where appropriate, his/her workload. At my unit, at the time, the radar could be prone to failure. And there were scheduled regular outages.
We took the strip scan extremely seriously.
As HD implied above, it's as basic (and important) a tool as keeping left on the road. Just takes a bit longer to learn.
And although, particularly the older controllers, view the discipline as foundational, I think that it is becoming less so, and radar (and other techy toys) the more basic foundation. In a few years I wouldn't be surprised if there aren't any controllers who know how to keep airspace safe using strips and a clock. (Heck, in a few more years, I don't know if controllers will be able to handle traffic at all using level and heading instructions, should the computers fail. But they won't fail, will they?
)
BTW, the clock is, IMO, important. Time appears to lapse in an analogue fashion. For keeping that 4D picture alive, there is nothing like having the time displayed in a pictorial fashion: big hands rotating around a big dial. Digitals just don't present reality realistically, to me.