I am new to SharePoint 2013 and I have been trying to find "step-by-step" tutorials on how to create sub sites programmatically and other ways of using SharePoint in programmatic manner?
I've been going through the same process as you but maybe a couple of weeks ahead...there are loads of good resources out there, mainly from MS. The best I've found so far has been the code examples.
You can filter by SP2013 and even the object model you want to use. I found the options in object models confusing at first, tried a few out with C#/VB then opted for JS/JSOM.
Also get yourself signed up for a developer 2013 site...the foundation version requires some amount of juice to run locally!
I follow blog of Chris O'Brien. He publish really great articles and code samples. This is not promotion, I really enjoy reading this blog)
Most of the content that you can find online, on books, communities & from past (& future presentations) is still relevant to SharePoint 2013.
Without knowing your specific requirement, it's a little complicate to guide you to a specific walkthrough but rather than being stuck with the rather low amount of dedicated content for SharePoint 2013, I heavily suggest that you search for similar requests with SharePoint 2010.
Creating site programmatically hasn't evolve from SharePoint 2010 so any good book (O'Reilly, APress, Wrox, etc.) will definitively help you to get things up & running with minimal amount of issues.