One thing to keep in mind -if you're using a software package such as Minitab, is what normality test they use by default. You can use several specific normality tests within Minnie, but it defaults to the A-D test for the basic reports. Just FYI. Also - I believe some of the tests are sample size sensitive, so you might take a look at how much data you have as an input to which test you select.
You can use a non-parametric test like Mood's median to see if any differences exist without worrying about distribution though, but take a look at what the data is. If it's something like a hole diameter then you might expect that to be normally distributed. However, if you're evaluation cycle times or arrival times, you would look more toward a lognormal, so a t-test might not be what you choose. Make sure you have continuous or interval data, not just numbers.
Cheers,
Adam