Best Way to combine multiple datasets into one model
Why don't you give Multi-label classification a try.you can actually train your model to predict multiple labels for one image by training the model on a dataset that has images with multiple labels the only problem you might face here is to where to get the data from you can actually use this dataset for practice:
https://www.kaggle.com/c/planet-understanding-the-amazon-from-space/data.

Another better option that you have is as you said in your question you want to train your model on multiple classes you can actually use Imagenet dataset it mas 1000 classes and more than a million images.
Note:Use tranfer learning to build your model and do watch lesson 1,2,3 from fast.ai part1 2019 course you will have all your answers. Jeremy Howard is a god!!!!