The most likely way to get sick is to surround yourself with people who already have contagious illnesses. Some people believe that going outside with wet hair will lead to colds, but this is actually a myth.
I would strongly advise against trying to make yourself sick, but that being said, here are some of the more elaborate ways I can think of to get sick.
Places like public toilets, buses, trains, and other areas with lots of people are riddled with all sorts of germs. I'm pretty confident that if you went around touching things in these areas and then ate a sandwich without washing your hands, you'd probably catch some sort of stomach bug.
This is extremely dangerous, so I would advise that you don't attempt it, but too much alcohol is often the culprit of feeling a little worse for wear. Of course, you could actually die if you drink too much, so it's not a wise choice.
Uncooked meats, especially chicken, can lead to very nasty bouts of food poisoning. You'd likely end up spending most of your time in the bathroom if you did get food poisoning. Again, this can lead to death if you're not careful, so I implore people not to try this deliberately.
Ultimately, it's not a very smart idea to try to make yourself ill. The most common way of getting sick is probably by spending time with people who are already ill. Other methods usually involve poor hygiene or questionable social behaviour.