If I need to engage a 150 or 200 yard target, how many "clicks" do I need to adjust for?
Depends on the scope and how much movement that scope gives per click.
Some are more granular than others. Basically you need to know how much MOA change there is per click. 1/4MOA is common.
You will also need to know the bullet drop for your round. If your bullet drops a nice even inch per 100 yards then it's one thing, if it drops 2 inches per 100 yards it would be another. And of course they don't drop in a nice even manner.
If you will look at photos of long distance shooters you'll see a laminated card hanging off the optic. Lots of research goes into making that card
Zak Smiths website probably tells how somewhere.
Basically you have to take your rifle, your optic and your ammo and figure it out for yourself or duplicate what someone else has done and hope its close. Depends on how close you need to be.
For something like an industry standard, 5.56 on an M4 for example you can buy scopes with built in BDC (bullet drop compensation). As long as your rifle and ammo meet the standards the BDC is set for you will be pretty close.
If you want the math:
At 100 yards:
Circumference of a circle with a 100 yards radius = 22,620 inches.
There are 360 degrees in a circle so:
One Degree = 22.620/360 = 62.83 inches
There are 60 Minutes in one Degree so:
One Minute = 62.83/60 = 1.047 inches
Luckily for most of us that .047" isn't critical so the 1" per 100 yard rule works well enough. 10.47" at 1000 yards.
That's why the scopes are more granular for some shooters. If you are shooting at 1000 yards a 1/4MOA click moves your point of impact a whopping 2.6 inches. That's waaaay too much for those guys so lots of optics have tighter movement, 1/8MOA or whatever.
Out past, what is it, 1500 yards? It's all way beyond any of my rifles ability (and probably mine too) but somewhere out there you have to begin to take into account the rotation of the earth in your trajectory. That's senior year stuff, I'm still around a sophomore I'm afraid.