There are a number of processes that you would end up working through in start up mode. In terms of relevance here are the ones that I worry about on a daily basis:
a) Product development - Take an idea from inception to launch. The core thought here is an iterative process that allows you to build - test - release - repeat so that your ideas get in front of the customer as quickly as possible and evolve with his reactions - not in isolation.
b) Linking product features to customer pains - While there is a bit of this that you would do in product development - the real usage of output from this process is pitching your product to customers and your business to investors. It forms the basis for your marketing and investor collateral.
c) Shipping - Release management ensures that the right version is in the right place at the right time and you don't ship something that could be classified as a "lemon"
d) Pricing - Also an iterative process that you need to determine how much money are you leaving on the table and what is the right time to increase your margins by raising your pricing
e) Hiring and Talent Management - Finding and convincing the right people to come work for you. Firing the wrong people after hiring them. Retaining talent once you have found it.
f) Cash management - Setting up suppliers to give you credit, making sure you get paid first and pay later.
g) Expectation Setting - No real process here other than being honest and upfront about what is likely to happen and then making it happen. Set these across the board with team members, vendors, suppliers, customers, investors and family.
There is a lot of material out there that you can read but my all time favorite are Founders at Work, Burn Rate and eBoys. A Google search on each of the above key words is also recommended.