Patching:
When patching smaller holes, a 6" taping knife should suffice.
Apply the compound in a few light coats rather than a single heavy coat, and sand between each coat.
You can either sand the patch using a sanding block like the 3M® Fine/Medium Large Drywall Sanding Sponge,
or wet sand using a slightly damp rag. If you choose to wet sand, let the area dry before applying the next coat.
For larger holes, you'll have to fill the hole with something solid before you apply compound (don't try to fill the hole with mud). Once the hole is filled, start with a 6" taping knife for the first coat. Next, apply a few thin coats using a 10" taping knife
feathering the compound out to blend into the wall.
When working with drywall compound it's a good idea to use a Hawk, or a mud pan instead of working directly from the bucket. This will insure that you don't end up with little bits of junk in the compound, and that the compound in the bucket won't dry out too much while you're working. They are also good for catching falling mud.
Preparing the wall:
Wipe the wall down with a dry rag, to remove any dust and dirt. Prime the whole wall with a proper primer, like Glidden Gripper.
Any time you patch, it's a good idea to prime the entire wall. If you simply prime the patches, you may be able to see the patches when you're done.
Painting:
Apply 2 coats of the desired paint, and you're done.
If the patches were done properly (smooth and blended appropriately), after primer and two coats of paint you'll never know there was a patch there.
NOTE: Products mentioned in this answer are for example only, I do not specifically endorse or recommend the use of any products mentioned.
Best Answer
Once it's been painted over, there's no way to remove a stain - you'd have to remove the paint to get at the stain, which is definitely not "minimal effort".
The simplest thing to do at this point would be to call back the contractor and have him actually paint the wall, rather than just slapping on some paint-like substance and calling it good. There is simply no way those marks would still be showing through if the contractor had done a proper job, with a stain-blocking primer and a good-quality paint.