I am glad you want to learn to do this type of job yourself. Before you jump in let's answer a couple of other questions first. How old is this place? If it was build before 1978, then you must by Federal EPA Law in all non owner occupied rental apartments test the peeling paint for lead before any scraping or sanding can be done. Simple test kits can be purchased at most HI stores, however it is the owners responsibility to have these tests done. If the test is positive for lead, then the work MUST be done by a licensed EPA Renovator with the training and tools to remediate the lead hazard and dispose of the wastes. There are no exemptions to this law in rental units and failure of the landlord to comply carries a $10,000 fine.
Now let us assume no lead is present and you can proceed with the job. The first step is going to scrape off any and all loose paint. Using a 3 or 4 inch stiff putty knife, carefully remove as much old loose paint as possible. Applying new paint over an unstable base will lift the old paint as you roll it on and make a mess. If the surface is too rough and uneven, then a skim coat of drywall compound will be needed if the entire wall surface needs to be leveled. If just small areas need leveling, then you can use spackle compound. If a complete skim coat is needed, the compound is best applied with a 12 inch knife about 1/16" thick over the entire wall. Smaller repairs with spackle can be done with a 4 or 6 inch knife. After the compound has dried, it has to be sanded smooth. Use 150 grit paper on a large sanding block. Do this gentily to remove ridges and give an overall smooth finish. You can go over it a second time if necessary with compound and finish sand with 220 grit to achieve a very smooth finish. Once you are happy with the smoothness of the wall, then prime it with a drywall sealer/primer, not regular primer. After you seal/prime, but before you paint, check the surfaces and make minor touchups using lightweight spackle.
There are several other tricks of the trade to help make a full skim coat go easier. Mix the compound well, add a little extra water and some Ivory dish soap as you mix. The idea is to make the compound a little thinner so it will be easy to spread evenly and minimize the tiny air pockets/bubbles. The larger the knife or trowel you can use will make the job go faster and leave less ridges. Do not overwork the compound or try to work it if starts to dry.
At this point you will be wondering why you agreed to this big job, but it will look great. So finally you can get to work painting with your selected color.
Best Answer
My bet is there's something in the old paint that's preventing good adhesion for the new paint. It's possible they used an oil-base indoors, or the paint had some chemical that messes with the latex you're likely using.
What I would do is buy an oil-based (mineral spirit cleanup) primer (like original Kilz) and prime the walls. Oil-based primers do a superb job of providing a "glue" layer for your latex to grab onto. Take a small area and test this hypothesis.
If that still fails, the next best thing is to put new drywall over the old and then paint fresh drywall. Obviously more expensive (and you'll need landlord permission), but you'd be guaranteed no paint problems.