You'll need to look straight on at the gaps around the door (top and sides) - any place you see daylight (like the bottom left corner in the first picture) will need to have the weather stripping replaced. Usually, this means replacing the weather stripping along that entire side - to my knowledge, piecing it together along one side probably isn't the best bet. If you have to replace some (and you do), you may as well replace it all. It probably won't cost must more to get a kit for the whole door than it would for one piece.
If brand new weather strip still doesn't prevent light coming through, you'll need either thicker weather stripping (to span the larger gap), or you'll need to adjust the door in its mount to better center it (that would be a whole new question here, feel free to ask it if you need to).
If there is light coming through between the wooden trim and the frame (i.e. not through the door opening itself), then you will need to apply caulk. A good quality exterior grade caulk would be appropriate. You might get one in a matching color, or you might get a good paintable caulk and choose to do some touch-up painting. You'll probably want to recaulk around the entire opening because "touching up" caulk is difficult to do and get it looking good, especially right there at the front door where all your house guests will see it. You'll want to do all the appropriate prep work for caulking as well - you should be able to find several good questions about that here, if not, we certainly need one!
If you see light at the bottom between the door and the threshold, you might be able to adjust the threshold, but I don't think yours is an adjustable one. Instead, you'd want to replace the door sweep on the bottom. The temptation would be to go with a big, thick, beefy one thinking "this will fill that gap!" but if it's too thick, it'll fill the gap so well your front door won't close. You will probably need to take the door off the hinges to get at the sweep at the bottom, so choose a day when it's not raining, or at least, not blowing in the front door.
Once you fix the seals around the door, then it won't get into the house to attack the moldings. Fixing up the current water damage would be grounds for a search here for additional info and maybe even another question.