Graphs can't tell you that a UFO has just spawned somewhere and you should go intercept it, for that they are not quick and granular enough... unless you are a complete autist (apologies to all real autists out there)...
Autist here. They're really not quick and granular enough. While a graph can tell you that a UFO has spawned in a region within a half hour, more often than not you don't even get a single pixel rise in activity while the UFO is merely flying through. This means that more often than not, you're not finding out about the UFO until it has arrived on site, and then the transit time of your interceptor can mean you miss the UFO. As Meridian said, it's better to just learn the two regions at the beginning of the month, and patrol them for the rest of the month.
Each month the aliens typically select two regions to begin a mission--one a terror mission and one random. As you shoot down terror scouts, you change the timing of the terror scout appearances, and same with other UFOs. Each one shot down delays its mission typically by some 2-6 days IIRC. Once you learn how each mission works, you can track it in the region it spawned in, and then patrol that region until it has completed. You can also stop patrolling for 2-3 days after shooting a UFO down for that mission, and focus elsewhere.
But I find that putting so much effort into the game is fun only a few times. After a while, it's just wasted time. I recommend using graphs to find a few UFOs in the early game to boost your research and money. Once you have that squared away, you can easily get by without touching the graphs, as long as you shoot down enough UFOs during the month and make enough points. There's several things you can do to increase how well you do in the game, but I recommend you just pick a couple and mostly ignore the rest.
A.) Check graphs at the beginning of the month to discover the location of the new random mission.
B.) Make multiple bases with interceptors and shoot down every UFO you can. (Make sure your bases are defended.)
C.) Run as many crash sites as possible.
D.) Always run every terror site, and also hunt down alien bases and run them. (Patrol with aircraft to find them. Radars won't see them.)
E.) Get Hyper-Wave Decoder ASAP and be sure to deal with every UFO that is on an infiltration or retaliation mission.
F.) Try to get at least a small amount of activity in as many regions as possible each month.
G.) Blaze through research and use research points to maintain your score.
H.) Do lots of manufacturing for money. (Vanilla: make laser cannons.) Make sure you put the money to use improving your score.
I.) Get fusion balls and destroy (not crash land) as many UFOs as possible (improved score over crash and forget).
My favorites are B, H, and I.
If your gameplay becomes too meticulous, consider that while your actions may be improving your game-time effectiveness, you might be getting less from the game in real time. You should decide how much time you actually want to spend playing the game, and be willing to let a strategy go if it's too tedious for your time constraints.
There's also varying ways to play on the battlescape:
A.) Basic rifles - shoot everything with direct damage. This method is clean and simple, possibly tedious but tends to have a high loot amount.
B.) Smoke for camouflage - put smoke down everywhere and use motion scanners to see through it. This method is very tedious but extremely effective.
C.) Explode everything - use rocket launchers, grenades, or blaster launchers to destroy everything that might have aliens hiding in it. This method is quick and dirty, great for scoring quick kills or completing a mission, but terrible for bringing back home any UFO components. On terror sites it's a quick way to get a neutral score.
D.) Save scum - save every turn or multiple times per turn, and reload the game every time something goes wrong. This method is the most tedious but results in a perfect score every time. Disadvantage: you won't be learning many of the important strategies that other playstyles will teach you.
E.) Stun everything - once you get the small launcher, just hit everything with stun bombs. This method is both quick and clean, though ammo can get expensive, and in a few cases it doesn't work well (some terrain or some enemies).
F.) Psi-hopping - Mind control every alien you see, and use them to spot more aliens. With effective unit placement, this strategy can strike with surgical precision. It can get tedious, however, and is only effective with strong psionic troops.
G.) Super soldier training - take extra time to train your soldiers' bravery, reactions, and any other attributes that tend to get left behind. With a lot of early work, you can get a team of soldiers with maxed-out attributes. This strategy is most effective in tandem with strategies that reduce risk to soldier life.
H.) Meat shields - Use tanks or rookies to draw fire so that your important soldiers are kept safe. You can also use the meat shields as scouts, and your main soldiers as snipers. This method is pretty effective relative to its low tedium, but relies on good terrain.
My favorite battlescape methods are A, F, and H.
