It's not "counter-intuitive" steering, it's "counter" steering.
You press down on the bar in the direction you want to turn, simple as that. It induces lean by turning the front wheel the opposite direction, which puts the weight towrds the turn. Press the outside bar to straighten up.
If you're too hot for the turn, just press more on the low bar. Do NOT brake unless you can sit it up straight. If that's not enough, you might get away with a touch of rear brake. After that, you just accept the fact that you screwed up and are going to crash.
Sorry for the length of the following. . . .
As for riding "safe" tips, basically pretend you're invisible. Make NO assumptions about whether you've been seen by other drivers. There's no such thing as a fender bender on a bike.
If you've never been on a motorcycle and someone says, "here, try it out," do NOT just grab gas and clutch and get out in traffic. We've all laughed our butts off at those Youtube vids where somebody gets on the bike for the first time and rides straight into a tree, a lake, a house, a car, whatever. It looks funny, you wonder how they could be so stupid, but a new rider with no muscle memory of the controls panics when the bike goes, and completely forgets that all he has to do is squeeze both levers to stop. Those vids are EXACTLY what happens when somebody just gets on and goes.
If you just bought your first bike, have an experienced rider bring the bike home and practice in your driveway. (My personal knowledge of a minimum mileage totalled bike is 7 miles on the odo!) First thing is to gear up and sit on the bike. Learn is the controls. Touch each lever and operate it with the bike off. Touch the brake. Hold in the clutch and operate the shifter. Find the choke. Find the petcock on the tank. (Fuel-injected bikes won't have either of those.)
Now it's time to learn the clutch. Sit the bike, raise the sidestand, turn the key on, and find neutral on the shifter; a green N light will come on. Start the bike with the clutch pulled in. With the clutch still pulled in kick the shifter down for 1st, and release the clutch just enough to feel some friction, but don't let the bike move. Feel how the lever moves to get the start of friction, then pull it back in without moving the bike. Slight release, pull it back in. Repeat several times, with just a little bit of gas. Bike clutches are wet clutches, so you're not gonna burn it up. Do this for 3 or 4 minutes.
Next, walk the bike forward while slipping the clutch. Don't put feet on pegs, just walk and then stop with the front brake. Walk and stop. Same thing, several minutes. What you're doing is building muscle memory for the clutch, gas, and brake.
Lastly, go ahead and ride forward. A little gas, release the clutch fully, feet on the pegs, and then pull in the clutch and stop. Use both brakes and put your left foot down first when the bike stops. Do that several times.
When you're comfortable with getting the bike moving, go ahead and get 2nd gear, then stop. To stop, pull in the clutch, apply brake, and downshift to first. Put your left foot down first when it stops. You should be in first by the time you stop, not stopped and then kick down. That's VERY VERY VERY important. Always downshift while still moving.
Once you've done all that, you've done the first exercises of the MSF basic rider course, and you're ready to start cornering exercises.
My point here is not to try to teach an MSF course, but to try to keep some noob from breaking his bike and his bones because he's "smarter" than the rest of us and was born knowing how to ride.
