Though it seems like a relatively simple task, collecting a water sample for analysis actually takes a bit of finesse if you want the testing to come out right, as in accurate. Anyone can do it, though, if they follow some simple instructions.
If possible, use a new, clean plastic container that you have rinsed out several times using the water to get tested. Make sure the lid gets rinsed off with sample water, as well. Take careful precautions NOT to touch the inside of the container or the lid with your hands, as that may result in contamination of the sample by oils or other compounds possibly on your hands.
Experts in the field of water sampling suggest shying away from the use of glass containers for sample collection when performing tests for metals but suggest using glass when testing for things like VOC's (volatile organic compounds).
When filling the container, do your best to ensure that no air, or as little as possible, exists in the sampling vessel once the cap gets put on... and when putting the cap on make sure its threads line up perfectly with the threads on the sample container. You should get a nice, tight seal that way and no water should escape the container.
Most home water testing can get performed with between 4 and 8 ounces of water, but certainly make sure you read the sampling instructions for the testing you intend to have performed since all tests require different volumes of water.
Try to get the sample packed up (carefully) and shipped to the lab as quickly as possible, meaning same day if possible. If that cannot happen then place the sample in a refrigerator for the overnight and ship as early as possible the next day. Definitely opt for an overnight shipping method and if called for by the testing you wish to have performed, include a chilled ice pack with the sample.
Clearly label your sample and make sure you include any required identifying paperwork called for the laboratory with the sample.