"Trained only" means they can only be used (you can only roll on them) if you have ranks on them.
There can be roleplaying reasons why your GM thinks you shouldn't have ranks on those skills, and you will have to work hard at them to be as good as with your class skills (due to the +3 trained bonus those have), but there's no real rules reason any character can't put ranks in any skill they chose (except for Fly if you don't have any means to fly).
According to the Rules As Written, you can put a skill rank in ANY skill, class skill or not.
http://paizo.com/pathfinderRPG/prd/usingSkills.html says:
n addition, each class has a number of favored skills, called class skills. It is easier for your character to become more proficient in these skills, as they represent part of his professional training and constant practice. You gain a +3 bonus on all class skills that you put ranks into.
So, with a non-class skill, you can still put in ranks, letting you try 'trained only' skills, but you don't get the +3 bonus for a class skill.
So your DM has either misinterpreted the rule, or has made a 'house rule' to use the skills in a slightly different fashion.
In my opinion, non-class skills should rather be encouraged than forbidden because they can bring a lot of flavor. If you look at the famous example of Lord of the Rings. A wizard can ride on a beautiful white horse, he just isn't going to be as skilled as a ranger or fighter that chooses to train in mounted combat. As for the trained only skills: Knowledge (religion) is trained only. I think this means that if you haven't studied undead creatures, you don't know their traits and abilities and can't roll for it. It can't mean that you are unable to read a book about it and gain the knowledge.
Also, if you want to give your GM a history lesson, version 3.0 of D&D had exclusive class skills that could be taken only by members of classes that had those skills as class skills. D&D 3.5 eliminated them, and Pathfinder certainly did not bring them back.
Important thing to note is that Knowledge skills aren't 100% trained only.
Untrained: You cannot make an untrained Knowledge check with a DC higher than 10. If you have access to an extensive library that covers a specific skill, this limit is removed. The time to make checks using a library, however, increases to 1d4 hours. Particularly complete libraries might even grant a bonus on Knowledge checks in the fields that they cover.
So, DC 10 questions about any knowledge can be answered by anyone without an int penalty as long as they aren't under duress or distracted. This represents things that would be considered "common knowledge".
Important thing to note is that Knowledge skills aren't 100% trained only.
Untrained: You cannot make an untrained Knowledge check with a DC higher than 10. If you have access to an extensive library that covers a specific skill, this limit is removed. The time to make checks using a library, however, increases to 1d4 hours. Particularly complete libraries might even grant a bonus on Knowledge checks in the fields that they cover.So, DC 10 questions about any knowledge can be answered by anyone without an int penalty as long as they aren't under duress or distracted. This represents things that would be considered "common knowledge".
Or 'easy Jeopardy questions'.