Newer veneer is usually pretty difficult to restore as the veneer layer is pretty thin. If you just have white water stains, this is damage to the finish and can be restored without doing anything to the wood. If you have black water stains, that is damage to the wood and is very difficult to repair on modern veneer without “burning through” the veneer when sanding. You could, of course, replace the veneer but this typically requires more equipment than most people have. Having more clamps than Norm is a start.
If you are in a situation where you will have people using your furniture that don’t take the care you do, plan for the worst. Solid wood furniture is more restorable. If you have young kids, furniture with a laminate surface will make life a lot easier and can look pretty good. If your decor allows, another alternative is tile topped tables.
Henry in MI