Can I use GCP for training only but predict with my own AI machine?
Decide if you want to use Tensorflow or Keras etc. Prepare scripts to train and save model, and another script to use it for prediction. 
It should be simple enough to use GCP for training and download the model to use on your machine. You can choose to use a high end machine (lot of memory, cores, GPU) on GCP. Training in distributed mode may be more complex. Then download the model and use it on local machine.
If you run into issues, post your scripts and ask another question.
My laptop had a problem with training big a dataset but not for predicting. Can I use Google Cloud Platform for training, only then export and download some sort of weights or model of that machine learning, so I can use it on my own laptop, and if so how to do it?