What are the immediate steps I should take when starting to bootstrap a new startup with a friend?

Building a little on Jeff's answer, I'll summarize what I've learned over the last year (in order of importance):

• Communications are critical; above all else, make sure you can ALWAYS be on the same page. Time is scarce, there's a lot to do, and you don't want to duplicate effort, waste time, or see stuff fall between the cracks.
• Create some sort of agreement on what each of your responsibilities are, how to agree on larger decisions (and what constitutes a big decision), and how you will handle revenue. The revenue part should be rough guidelines on how much profit to take, when, and whether you use some sort of ratio to compensate one of you over the other (if applicable)
• Define a timeframe where you agree to stick to this agreement without (major) change - it's all there, black and white, and (like #1) you'll be on the same page and can worry about other stuff. An agreement ain't much use if one of you wants to change it all the time.
• Get some help for legal and accounting stuff - unless you're a pro at either of these, it's just something you won't be able to handle. This is NECESSARY investment. It's not a ton of work, but you'll need to get some paperwork done to be able to get a company bank account and such, and make sure you're following IRS rules.

Getting this far together should give you a good idea of whether you should go forward as a team. A vision and idea for a product is important, but you can and will probably change or evolve these over time. At least having completed the steps above, you can go forward confident that you have a solid foundation on which to build your company and can focus on creating a compelling product.

Generate lists of concerns and topics to discuss and set a time to cover as much as you can. You consider this a bootstrapped project, but as soon as any money comes in, someone is going to want to spend it. Make sure you plan that before euphoria sets.

Determine your roles. You want the company to be 50/50 but if one of you has an area of expertise, maybe their opinion should hold a little more weight in that area. Decide this ahead of time.

Once you know your role, you can make a commitment to how much time and energy each of you can contribute. Decide this ahead of time.

This process shouldn't drag on too long. If it does, maybe it wasn't meant to be. You don't want to get caught up in working on every detail if it's going to hold the project back. Not doing it kills more startups than anything.

The term "Bootstrapped" essentially is a play off Eric Ries' Lean Startup Methodology. And if you haven't watched Eric Ries' “Don’t Be In A Rush To Get Big, Be In A Rush To Have A Great Product” roughly 15min promo video of his new book on Tech Crunch, I'd highly highly recommend watching it right now: http://techcrunch.com/2011/08/14/lean-startup-eric-ries-tctv/

Moving forward with the assumption that you are truly going "Bootstrapped" and "Lean," any time you spend outside of perfecting your product will be an inefficient use of your time.

You can read and digest for days, weeks, or even months on topics of: fair co-founder equity shares, figuring out what development platform your web application should rest on, motivating and retaining your employees, social media strategies, breaking down the cost of your startup, what type of legal entity you should form, what Angels and VC's are looking, how to make an awesome Angel.co profile, etc, etc, etc..

Before you get lost in the clutter of it all, let's get you started with the following:

1. Perfect your product to market fit, make sure people really want what you're offering. Interview people who are NOT your friends and will give you candid feedback. How bad is their pain point? How great is your value proposition?
2. Innovative disruptively. If the market is saturated, your execution has to be remarkable!
3. Develop a super super Minimal Viable Product
4. Gain traction. And then evaluate your next step

Good luck!