It really does depend on what is in your water, and how much is there. You can buy home test kits to find out, at least roughly, or you can bring in one of the many water treatment companies. (I'd do the test kit myself first, to see if they are telling the truth anyway.)
Assuming you do have iron in the water (manganese has similar chemistry, and probably is taken out the same way.)
Iron/manganese can be removed in several ways. Again, the best scheme really depends on how much is in the water. For example, if there is just a bit of iron, along with other stuff, then a softener might be a good choice. To get a lot of iron out, you may need to go all out with a chlorination system, then filters to remove the remaining chlorine.
As far as cost goes, if the iron is actually leaching out of old pipes in your own walls, that is a major cost of course. Tearing out walls and floors to replace plumbing will cost you large sums of money.
If a chlorination system is used, coupled with filters, a settling tank, and a chlorine pump, then expect a cost of at least $3000 to $5000. (Depending on where you live, it may be more.)
Smaller problems can cost less. If only a water softener unit is needed, then I'd expect the cost to be somewhere in the range of $500 to %2500.
If all you need is an inline whole house iron filter, that can be installed by a handy person with a torch and bought from the local home center, now you are talking several hundred dollars.