Since you said rubber bumper, I am assuming it is a urethane like most bumper covers today are. I use a 2 part material for use on urethane products. Check through the products made by companies such as fusor, duramix, usc, ect. Lots of companies make stuff for the repair, A decent auto body supply store should have some. You may need a special gun to apply larger tubes of material, but smaller amount they sell in a dispensor packaging to mix the two materials. I've had good luck with the small bondo flexible bumper epoxy repair material that comes with two tubes you squeeze equal amounts of and mix together. About the cheapest way to go if you only need a small amount of material for repair. Some people use a plastic welder and urethane rod, but see no real advantage using it or buying a airless welder if you don't have one. Urethane is thermoset and you are relying on the adhesive bond and won't be melting the urethane.
You will want to chamfer around the hole a little and grind an inch or so aways past the hole on both sides of the bumper and sand with around 80-180 so you have bare urethane aways past the whole and a good featheredge on the side of the bumper you see. Basically the bigger the hole, them more area you will have grind sand and feather edge for a good repair. Apply the adhesive with a spreader pushing into the hole and build up the material a little bit, and spread and blend out past the hole. Build up and apply some more to the backside for aditional support. If this is a bigger hole, you may have to due one side at a time and tape the backside so you can get material where you need it, then grind off the tape and do the other side. You can sand the repair material with 80 grit on a block or dual action sander, and do finishing work with a thin swipe of plastic body filler or 2 part spot putty. Go over it all with 180 on a da before priming. Basically you don't want to just plug the hole with repair material, you want a little more surface for the adhesive to hold to. The scratches you can fill in with the repair material after sanding around them and sand off, fill with 2k spot putty, or maybe just featheredge out and prime. If they are actually cracks, then they will need to be v'd out and repaired a lot like the hole. Some will spray an adhesion promotor on bare urethane before priming for extra assurance. I've done both ways with and without, and both seemed to work fine for me.