The best method is a tiered combination of methods. Under the assumption the colonists are trying to find a suitable food source before their own stores run out a tiered approach is best.
Common sense, don't eat scavengers or parasites (both mobile and not), be wary of anything immobile, don't eat anything that is dramatically colored compared to other wildlife. These are all things that make parasites and toxins more likely.
Chemical analysis to make sure no obvious toxins are present, common elements on one planet might not be on others, arsenic might be widespread. The local wildlife will have evolved with it and thus it will be harmless to them but toxic to humans. You will also want to do this for nutritional information, eating all the local wildlife you want will not help if none of the local wildlife has any vitamin B, or only has lefthanded nucleic acids. You also need to make sure they use molecules of the same handedness, if not your colonist will be better off commiting suicide.
Animal testing, rats or dogs for cost, chimps if cost is not an issue. Of course if you have lots of earth animals you should be eating those instead. If you don't have earth livestock, Start with human gut cultures and cloned tissue instead, otherwise these will be the next step. Consider starting with products fermented by earth bacteria, alien product/earth bacteria cheese is one of the few things with a decent chance to be edible.
Cook everything, many parasites and toxins will be destroyed by cooking. It also means they need to eat less of the alien food (cooked food has much higher nutrient availability) and thus reduces chances of an adverse occurrence.
The universal edibility test, Timing is important in this test, and it checks for things like allergic reaction and acute effects earlier tests will miss. You will almost certainly lose people at this stage. Allergic reaction is going to be a big worry since there is a large risk of novel molecules. This is something that can only be tested directly on humans, even tissue exposure test will not catch many allergies, so including this step will be important. Each human will have to do each test for each food but you don't want multiple people testing the same food at the same time. preferably you want days separating them. There will be quite a bit of conflict here as they decide who the most disposable crewmate is. Note you are going to want to add another few steps to this, first prior to injection a pin prick test will be useful. during ingestion testing you will want to start with minute quantities and work your way up to substantial amounts.
All combined you are looking at months before anyone actually eats anything, and depending on the number of colonists years before everyone is eating it.