Here are the steps I'd recommend:
1. Pick out a dozen startups whose products/services REALLY interest you.
2. Formulate a couple of interesting thoughts/questions/ideas about each of them. This link provides some general ones to ask. You should also think about a few very specific to the startup's product/service.
3. Try to reach out to people who work at the startup through a variety of ways. For e.g. through shared contacts, an alma mater in common, writing brief inmails on Linkedin, or sharing your ideas with @theirtwitterhandle and getting to know their community manager, etc.
4. Learn from these discussions.
5. Repeat with more startups.
This is more of something you have to research first to find a startup. You will have to find a startup you like, and then reach out to them to see if they are actually hiring. It might be through Facebook, Twitter, their main website, or via phone.
The big thing to remember is to make sure you let them know all your credentials and what you can bring to the table for that startup. They don't want just a number, but someone that will be a key component to making sure they don't flop.
Make sure to talk about their business and some ideas you have to improve them. This shows your true interest in their business.
Don't come across as cocky and a know-it-all. You want to be friendly and be someone that they will actually want to work with.