Is there kind of a checklist what to consider before going on a day hike? Like which preparation should one take, things to pack (not detailed, like "food"), useful information (weather forecast e.g.) and so on.
It depends on your destination. In general I would think of:
Thinks to pack
Edited to incorporate comments
As well as checking the weather forecast you should research the general weather conditions for the area you will be visiting, often conditions in hilly or mountainous areas can change quickly and without warning so you should be prepared for the worst possible conditions you might encounter. This also includes checking sunset times for the time of year.
It is also important to consider your route carefully, looking at gradients and terrain as well as distance. Satisfy yourself that all of your party can manage it in the time available. Ideally you should have an alternative or exit route in case you are delayed for any reason. Even on well marked paths a map and compass and at least a basic understanding of how to use them are essential, GPS etc should be considered useful additions not the sole means of navigation.
Also consider any potential hazards on or near the route such as cliffs or rivers which could be danger if you get lost and/or visibility deteriorates. Some areas may have particular hazards in the form of wildlife or be prone flooding etc etc.
Inform somebody reliable of your route and the time you expect to be back and check in with them when you return.
In many areas it is wise to take at least some preparations for an emergency overnight stay. In most cases this will not be full camping kit but might include extra food and water, some sort of emergency shelter and torches. The simple fact of being prepared can do a lot to prevent panic if you do get lost.
Have an emergency plan in case you get lost or separated. In most cases this means deciding in advance whether people will stay put once they realise they are lost or head to a prearranged meeting point.
As well as the obvious safety implications this sort of thinking ahead will tend to make a trip more relaxed and enjoyable.
Some good answers here, but so far no-one has mentioned perhaps the single most important safety-factor:
Depending on the area and the season, the risk factors you should be considering might include:
Over the years, most of the issues I've seen in the hills have stemmed from a party underestimating the route or overestimating their skills.
Sometimes this can be quite extreme: I once had to help rescue a lady who was attempting Ben Nevis in full winter conditions and a gathering blizzard while shod in high-heel shoes!
So do your homework, get a realistic idea of what you're up against, and make a realistic assessment of whether it's the right route for you and your party.
I'll second the earlier nomination of the "ten essentials" list (@Nate Eldredge). There are two versions of this list, with the classic version of the list tracing its roots back to the 1930's. These lists are presented convenently on REI's web site (http://www.rei.com/learn/expert-advice/ten-essentials.html), but I will reproduce them below for posterity.
One bit of wisdom often not given in association with these lists is "...and the knowledge to use them properly". Without knowledge, these items are little more than additional pack weight. If you are going into any sort of a remote area, get proper training in navigation, first aid, and basic survival skills.
For anything much more than a short outing at a local park, I will pack enough for an overnight including all of the items on the modern version of the list. Understand that this would NOT be a comfortable overnight, but I would be protected from the forecast weather conditions and have sufficient food and water. This typically weighs in at 12-14 lbs (and this is where the ultralight backpackers get to chuckle...).
Updated Ten Essential "Systems"
Classic Ten Essentials