1. Play mid and jungle. Understand their flow and timing. I don't get ganked often by Shaco players because I play Shaco jungle. I know if we have a slow jungler vs. Shaco, he'll probably go from his Blue to our Red. I know to ward our Red, or if I'm playing mid, stay closer to the other side of the river. If I'm playing against an Eve, I know to back off around level 3 and level 6 if her icon is missing from the minimap, because that's when she likes to gank.
2. MIA calls go the same way; if I'm playing vs. Twisted Fate and he pushes into my turret, I call MIA immediately and tell side lanes to back off. As a champ like Eve, abuse this. Go invisible often. Force your opponent to call MIA a lot. Make him cry wolf.
3. Look at the direction a player moves before he leaves fog. The direction he faces and moves is telling for where he intends to go. I rarely ping a champion just once. I ping a projected path (i.e., 3 pings along the river from mid to bot). It's much more useful for allies.
4. Intuition is a strong thing. You should never say "I knew they'd be there." Good players don't get caught. They don't go somewhere knowing there's a risk an enemy team would be there.
5. While in fog, look for jerky and sudden movements from enemies. Do they respond to your being there?
6. Against players who buy a lot of wards, click them immediately to count their wards. When they head away from lane or towards brush, check to see if their ward count drops. Most players make it extremely obvious that they're warding. I don't--when I'm mid for example, I'll try to dip into the brush several times, or drop my ward while I'm naturally facing the brush. Abuse your opponent's inability to multitask--perform important actions while he's trying to last hit.
Build a habit of checking the minimap whenever particular events occur. For example, if you're support you could choose to check the minimap whenever a wave approaches, whenever the enemy dips into the bush, etc. Basically just make yourself a program where you train your eyes to regularly check the world.
I realized that I perform almost all navigation via the minimap. When I'm traveling between lanes, walking in the jungle... basically, any non-combat maneuver, I'll actually look at and click on the minimap to move. Jungling is a great time to keep your eyes on the minimap too.
I found that I trained myself to look at the minimap more by picking a jungler who didn't need to target the enemy with his abilities.
Then I spent 20% of my time looking at my character and 80% of my time reading the minimap for 10 games, even in team fights.
That exercise helped me a lot.