Training Deep Nets on an Ordinary Laptop
Yes, a laptop will work just fine for getting acquainted with some deep learning projects:

You can pick a smallish deep learning problem and gain some tractable insight using a laptop so give it a try.

The Theano project has a set of tutorials on digit recognition that I've played with and moded on a laptop.

Tensorflow also has a set of tutorials.

I let some of the longer runs go overnight, but nothing was intractable.

You might also consider availing yourself of AWS or one of the other cloud services.  For 20-30 dollars you can perform some of the bigger calculations in the cloud on some sort of elastic computing node. The secondary advantage is that you can also list AWS or other cloud services as skill on your resume also :-)

Hope this helps!