What is the best way to buy out your business partner?

There is no right or wrong way to do this.

1) Valuation--valuing a company, especially a new, small, or privately held one, is an art, not a science. A CPA can  help you--he or she may be able to do it him- or herself, or recommend you to a consultant. In brief, the most common valuations are:

a) Book value, especially for money-losing companies (or at least which aren't showing a profit): add the value of all physical assets, intangible assets (e.g. intellectual property) and accounts receivable, subtract all liabilities/debts, including accounts payable, and what's left is the value.

b) A multiple of gross receipts; every industry has different parameters, but for small, unproven companies, it's usually or 1 or 2 times  gross yearly receipts (or sales), and pretty much never more than 3 times.

c) A multiple of EBITDA, which is most appropriate if there is EBITDA, which not all new companies have. This varies widely by industry (so you need to see what applies to yours); for example, in educational print publishing, where  I used to work, common multiples were from 7x EBITDA to 12x.

At the end of the day though, it has to be price you both are comfortable with.

2) In terms of structuring the deal, you can do it anyway that mutually works. Cash up front, promissory note, percentage of earnings for some period of time, a mix of all three...however makes sense is legal, though obviously each has different tax ramifications.