I'd like to make an app using iOS's new CoreML framework that does image recognition. To do so I'd probably have to train my own model, and I'm wondering exactly how much data and compute power it would require. Is it something I could feasibly accomplish on an dual core i5 Macbook Pro using Google Images for source data or would it be much more involved?
It depends on what sort of images you want to train your model to recognize.
What is often done is fine-tuning an existing model. You take a pretrained version of Inception-v3 (let's say) and then replace the final layer with your own. You train this last layer on your own images.
You still need a fair number of training images (a few 100 per category, but more is better) but you can do this on your MacBook Pro in anywhere between 30 minutes to a few hours.
TensorFlow comes with a script that makes it really easy to do this. Keras has a great blog post on how to do this. I used the TensorFlow script to re-train Inception-v3 to tell apart my two cats, from 50 or so images of each cat.
If you want to train from scratch you probably want to do this in the cloud using AWS, Google's Cloud ML Engine, or something easy like FloydHub.