1: testing your time series follows Random Walk process:
a. Download your data(finance.yahoo.com), then you need to do the natural log transformation of your data, i.e. Yt=In(Pt): empirical study shows only the log price might follow random walk process.
b. Unit root test can only test whether the series is stationary or not. to testing random walk you need to test the error term is iid(independent identically distributed).
because if Yt follows random walk, Y(t)=Y(t-1)+e(t), then e(t)=Y(t)-Y(t-1)--the random walk process restricts the coefficient of Y(t-1) to be "1", therefore e(t)=Y(t)-Y(t-1), is the log return.
c. Generate your log return series(et), and test it is iid: you can use "Run Test", "Variance Ratio Test", or other relevant tests
2. Testing the market is efficient: among the three forms of Market efficiency(weak, semi-strong, and strong form), weak form is most relevant in this case. Because it argues that the current price incorporates past (public) information, to test it is equivalent to testing the log return follows a random walk process.