Before that, see what kind of time series you have. The unit root tests are valid if the time series is well characterized by an AR(1) with white noise errors. Many financial time series have a more complicated beahviour. Said and Dickey (1984) improved the unit root test to deal with general ARMA( p, q) models, with unknown orders, Their test is known as the augmented Dickey-Fuller (ADF) test.

It depends what you want to do. Many macro series increase exponantially, including  most price series. So, we take logarith to linearize them. Plot your series and see what they look like. Standard unit root tests assumes a linear structure. I suggest you investigate plot of your series carefully before you do any test to determine the time series properties of them. This would also allow you to see the shifts, breaks, outliers, etc. in your series.

Dear, if you are aware ARDL and JJ cointegration test so it is better to run OLS first to check the relationship because if series are in same order integrated so JJ is appropriate and if some series are integrated at level and some are at first difference so ARDL cointegration is better one.  

However, when we take log of CPI it will turn commodity prices into inflation rate. If you are focused on prices so do not take LOG on commodity prices. 

Here is an example which shows the usefulness of taking logs. Consider the stochastic process (s.p.) Χ, which is the product of a simple function of time (t), say exp(t), and of a stationary s.p., Z:

Χ = [exp(t)]*Z, where Z > 0, Z ~ i.i.d. (μ, σ2).   (1)

Obviously, the s.p. Χ is not stationary, since its mean and its variance both depend on t: Ε(Χ) = Ε[exp(t)*Z] = exp(t)*Ε(Z) = μ*exp(t) and Var(Χ) = Var[exp(t)*Z] = exp(2t)*Var(Z) = σ2*exp(2t). It is also obvious that taking first differences does not produce a stationary s.p. The log of X, however, say Y, i.e.,

Y = ln(Χ) = t + ln(Ζ),   (2)

although it is still not stationary with respect to the mean, since Ε(Y) = t + Ε[ln(Ζ)] depends on t, it is nevertheless stationary with respect to its variance, since Var(Y) = 0 + Var[ln(Ζ)], which does not depend on t, since Ζ is assumed to be stationary. The first difference of Y, ΔY ≈ 1 + gz, where gz is the percentage change in Z and where the approximation is valid for only small changes in Z, is obviously stationary with respect to both mean and variance.

Since the hypothesis "At most one cointegration vector" is rejected at the 5% level, the conclusion is indeed that you have two coin. vectors. With only two variables in the system, and assuming that you take this test more seriously than the unit-root tests, the only reasonable interpretation is that both of your variables are in fact I(0), despite the evidence from the unit-root tests. The reason is that each I(0) variable by itself constitues a coin. vector.

It seems to me that you test for the law of one price for the same commodity between two locations, say i and j. If so, you need to test the ratio of prices, p[i,j,t]=p[i,t]/p[j,t] for stationarity. The reason for taking log is in that this ratio as a function of p[j,t] (given fixed p[i,t]) is asymmetric about 1: values < 1 are confined in [0, 1) and values >1 are confined in (1, infinity). And log(p[i,j,t]) is symmetric about 0.

I guess you need not run cointegration tests; testing p[i,j,t] for stationarity is equivalent to testing for cointegration with predetermined cointegraing vector (1, -1). The test would be with no constant if the strict law of one price is assumed; if there may be a constant (percentage) difference in prices (due to e.g. transaction costs), then the unit rot test is run with constant term. 

If your work indeed relates to the law of one price, my paper Price convergence and market integration in Russia. Regional Science and Urban Economics, 41 (2), 160-172, 2011 could be interesting for you.

I suppose you need to perform a regression. If it is the case, you must test the datas in the same format as they are used in the regression. One of the best test to check the stationarity is the Augmented Dickey Fuller. If the test not holds, you have two other possibilities. The first is to perform in any case the regression, but without autoregressive terms, and then check the stationarity of the residuals (this is equivalent to perform a cointegration test). If the test not holds again, you have to transform the format of the datas. The most useful transformation is the first difference of logarithms, but sometimes it is necessary a more high degree of differentiation. Then you can perform the regression (without autoregressive terms) and check the stationarity of the residuals again. If also this second step fails, there is a great probability that the phenomenon that you are studying is per se non-stationary.