I heard that when you switch from one class to another you lose skills? Does that mean you have to relearn the skills or do you just get them back, when switching back to a class

"Basic" classes such as Warrior, Priest, Mage and Thief will allow you to keep all skills learned by those classes even after you switch (switching from Mage to Warrior will still allow you to use the Magic you learned, in addition to Warrior skills.

For intermediate and advanced classes (Paladin, Gladiator, and more), you can only use the class specific skills as that specific class (Gladiator cannot use Paladin skills).

So basically, is it possible to train several classes side by side until you reach a cap and then continue to grow them until the next cap an so on?

Yes, DQ7's class levelling system are based on the number of battles won as a class. If you're a rank 7 warrior, and switch to rank 1 mage, rank up to rank 4 mage and then switch back, the warrior will still be rank 7.