I'm happy to share our process around this. We started by reading a lot of examples of others' Guiding Principles (which include core values and a mission statement), thinking hard about what we want to do and talking about it at the board level. I then constructed some preliminary concepts and spend individually or in pairs with everyone on the team, getting buy-in and feedback. Eventually, we came up with a set of values we could all rally around (which was made much more challenging than if we had done it when there were only the couple cofounders, but a necessary process).
I detailed a bit more about the results of that process here - http://www.seomoz.org/blog/establishing-guiding-principles-at-seomoz I have to say, though, when you're very early in a startup, be ready to get flexible about mission and core purpose. We've evolved 5-6 times over the last 6 years and every time it's been good. Part of me is almost happy we took our time getting to a slightly more mature stage before adopting these.
Core values come from the tone you set and whether you practise them consistently....
Mission for a startup can be based on "What were we thinking when we were raising money" ex: We want to build the best underwater fire fighter in the world.
The mission could evolve / change with our business
Core values - These are irrespective of the business, product etc....These do not change....these are underlying pillars.....
ex: do not lie
ex: the ten commandments
ex: First and fore most we are responsible for our patients, doctors, nurses, mothers, and fathers.....This is not verbose but something similar from Johnson and Johnson...
The standard trifecta are the: Mission, Vision and Values statements. These are the traditional ways to communicate your culture. The one that I think is a lot better comes from Guy Kawasaki. It's the Mantra.
Most people don't remember the Mission, Vision and Values but they can remember a Mantra. The Nike example above shows their Vision and Mission, which most of us did not even know but we do know Nike's Mantra: Just Do It!
Along with the Mantra, you have to set the example for the kind of culture you want. Walk the walk, talk the talk and live the culture. No words on a page can replace that.
We put together an article about writing mission & vision statements years ago. The underlying idea is that these statements express the DNA of your company and help create a unified direction that everyone in the company can follow.
Vision: this is a powerful, big picture statement summarized into a single phrase or sentence.
Mission: sets out what you are going to do to accomplish your vision.
Here is an example from one of the world's brand leaders:
Nike Vision: "To bring inspiration and innovation to every athlete in the world "
Nike Mission: "Innovate for a better world "
I think it's important to be authenitic when developing these statements. Sometimes people will project what they would like to be onto their mission & vision statements, but that may not necessarily reflect who they are likely to be. When you are done these statements should resonate your unique DNA.