I have some experience with this.

Some "facts" just set my spidey-sense tingling, and I find it usually well worth the time to check out the references in such case. In general, with the slightest doubt I will at least Google the reference and check out the abstract - this is quick and will at the least guarantee that the source does exist.

Particular things that set my spidey-sense off are:

• sensationalistic claims - any that is ostensibly used to shock the reader into action
• excess precision - like claims about "56% of all software projects" vs "roughly 60% in the population we sampled"
• excess confidence about hard-to-test claims, in software those tend to be "productivity" claims
• claims that are ambiguous or that would be very hard to confirm experimentally, e.g. the well-known canard about how many times a day men think about sex; basically "is this even a sane question to ask"
• hard-to-find primary sources - when you can't readily check it, a claim becomes more suspicious
• abstracts that don't contain the claim being put forward - it's more suspicious when you cite a paper for a tiny bit buried deep within
• (ETA) "phone game" claims - a citation to a citation of a citation, etc. with many indirections before reaching the primary source

Let's look at some specifics of the page you cite - at the outset we note that it's designed to be sensationalistic, a marketing brochure basically. It's up to you to factor that into your assessment of how much you trust the references.

• "As many as 98,000 people die in hospitals each year as a result of medical errors" - doesn't feel implausible, but a quick Google for US annual death rate - it turns out to be twice the death rate for suicide. This document seems to contradict the finding, I'd check out the reference
• "Doctors spend an average of only ten minutes with patients" - an average like that isn't too hard to work out from a survey and squares with personal experience, I'd take it at face value
• "By some estimates, deaths caused by medicine itself total 225,000 per year" - excess precision for a phrase as vague as "caused by medicine itself", I'd check out the reference just to know what that's supposed to mean
• "Most published research findings are false" - this is the title of the Ioannidis article, and should not be taken at face value to mean "all research in all fields of medicine", read the ref for sure
• "Up to 30% of patients who die in hospitals have important diseases or lesions that are not discovered until after death" - I'd want to know more about how they estimate this - are we talking about extrapolation from the few deaths which result in autopsy?
• "It takes an average of 17 years for scientific research to enter clinical practice" - uh, maybe; somewhat ambiguous (what's an objective criterion for "entering clinical practice"?)
• "In oncology alone, 90% of preclinical cancer studies could not be replicated." - I get into trouble almost immediately trying to check this reference, given as " Begley, C. G. (n.d.). Preclinical cancer research, 8–10.": Google gives me partial matches on the title but no exact matches, (n.d.) means "No date" which is kind of weird; this does have a publication date and even a URL
• "deaths from cancer have barely been touched" - I wouldn't be surprised, cancer is a tough bastard
• "If a primary care physician provided all recommended [...] care [...] he would need to work 21.7 hours a day" - excess precision (also a hypothetical, so not really "evidence"); check the source

Also, this is a Web page, so I get suspicious on principle that no hyperlinks are provided.