What is a good way to push with your bumper without ruining the body of the car you are pushing (paint scratches do not matter.
The best option I have seen is to put an old tire (without a rim) between the cars. The tire is flexible and provides a large surface area to minimize the possibility of dents and scratches.
That said, many, many cars these days are just not pushable or capable of pushing. Many "bumpers" are for looks only and won't stand up to any real force without breaking attaching clips or internal mounting brackets. My general rule would be that unless it has a chrome, metal bumper or you know for a fact exactly how the bumper is constructed or attached to the car, DON'T PUSH IT.
Assuming you don't do anything really stupid the worst case scenario is the bumper of the dead vehicle riding up over the bumper of the push vehicle. I have have broken more than one grill doing this. But the grill is not the big issue, you also have the radiator and other fragile parts just behind the grill.
If you are hanging around the same crowd I did back in the day, there is more than one potential push vehicle available, the best choice is one where the center of the bumpers match of perfectly. Second best is the push vehicle being a bit higher. A bit lower is bad (per above). You can sometimes lower the back of the dead vehicle by putting junk in the trunk. Concrete blocks, engine block, transmission, etc. Optimally if the one bumper slides on top of another you want the good vehicle to get the high position.
Cornering and bumps (onto driveway from road) are the two biggest problem areas. I you are pushing around a corner or over a bump that is when you tend to break, head lights, tail lights and get the bumpers on top of each other. The push vehicle should provide enough energy prior to the obstacle to get around the corner, or over the bump then fall back. Once past the obstacle the push vehicle re-contacts the dead vehicle.
In all cases when touching bumpers you want relative speed to be close to 1 mph. If the dead car is stopped, move forward slowly until they touch. If the dead car is still rolling, ease into as slowly as possible. When they touch give the push vehicle a bit more power, you want to keep the pressure on, if you don't push a bit harder on contact you will bounce back then ram the dead car when you accelerate. I use both feet for this, one brake, one gas. It takes a little practice, but it is easier to learn than keeping a tow rope tight when flat towing.
Some people like to use a tire (without a rim) as between the bumpers of the two cars. You can tie it to either vehicle, it doesn't really mater. In my experience this just tends to put black scuff marks all over. Back in the day of chrome bumpers it had value, but since around the 1980's bumper have had built in shock absorber and plastic/rubber covers. So not a lot of added value.
If you do have to push where the push vehicle has the lower bumper the tire can be a good choice to protect the grill and other fragile parts.