[Pillsbury]

Hi Ana Laura,
Sorry about all the capitals, my italics isn`t working. I`m offering a strategy to help with this scenario in general (when we have words of the same meaning to teach) and will use your words for examples. Part of what I offer is already in Susan 53`s response.
First, having a good English to English grammar dictionary is handy. I suggest you get one if you don`t already have one.
Then, the strategy I would use is to concentrate on ONE aspect of each word`s meaning, (keeping them as different as possible.)
For example:

GRIP: Teach it as a noun that can be used as a verb. (Farm is an example of this. I know the other words Grab and Grasp could be explained similarly but the stress is the best way to use each word. )

Susan 53 is right in mentioning idiomatic expressions, but when we try and explain this we get into a quagmire of how all the grammar rules change when dealing with context. Therefore teach sayings and idioms as separate instances, not meanings. i.e. "This is a SAYING/IDIOM using the word 'X' and this SAYING/IDIOM means 'Y'", not, "This is a USE for the word 'X' and it`s meaning."

A movie can be 'gripping', but what that MEANS is that the movie is 'compelling' in some way. We 'look for' things, but what that MEANS is we are 'searching.'

When used as a verb, 'Grip' literally has to do with the strength of the hold applied by the hand. As a noun it is also something for the hand to hold onto, i.e. a bike grip on it`s handle. Grip is also a function of things, i.e. Tyres grip the road because they 'have grip.'
As above Susan 53`s examples, "She gripped the chair,"(means "she had a grip on the chair." She could not have 'a grab' or 'a grasp'. If she had 'a grasp' it would mean she 'understood' the chair.) Instead, "She has a grip on it."

She could 'grasp' or 'grab' the chair...read on.

Grab: Is fast, and as Susan 53 says it can be fairly desperate. However I would teach it as careless or reflexive. Also as an act meaning to acquire something/to snatch.
"He 'grabbed'(snatched) his car keys off the coffee table as he left in a hurry", "She 'grabbed'(snatched) the toy from the other child", "He grabbed(snatched) the rope because he was falling." In all of these instances, the intent is ACQUISITION.
Grasp would sound a little out of place in these situations if used instead of grab.
In the grip example, "She gripped the chair," one could grab or grasp the chair, but there would have to be movement. "She grabbed/grasped the chair [because she or it was falling]."
In grabbing, they are SAVING IT OR THEMSELVES, in grasping they are MAINTAINING POSESSION.

Grasp: Here is where I would use desperation. If we grab a rope there is no certainty we are holding it, if we grasp it, we most certainly are holding it. Strange but true. You can not grab with more than your hand, but you can grasp with more than your hand. Grasping also implies use of the arms and body. In this sense, it`s akin to hugging/holding very tightly AND close.
"He grabbed his car keys/rope," implies HIS movement as he grabs his car keys/the rope with a hand. "He grasped his car keys/the rope," implies that he ALREADY HAS his keys/the rope AND is HOLDING VERY TIGHTLY AND CLOSE, as in, "She grasped the doll to her chest."
"She grabbed the doll to her chest," is used, but is not good grammar and is slightly out of place because it means, "She grabbed the doll [AND THEN] grasped/hugged it close to her chest."

Likewise with the word grip, "She gripped the doll." The meaning would have to do with the strength applied by the hand, not the action itself. (The tightness, not the holding.) The meaning is more subtle and psychological. Gripping the doll, the child is squeezing with their hands; grasping the doll, they are concerned with MAINTAINING POSESSION rather than the power they are using. When grasping, they may not even be using their hands at all, they may be hugging.

One could, "grasp [at/for]" something as one could "grab [at/for]" something, but that then is different grammar as well as idiomatic and a subject for another time once the basic meanings have been discovered and mastered by the class.
I hope I was some help. Good luck.