According to the company that installed my well pump and pressure tank, they say that this thing happens. Apparently some bacteria and iron are producing stuff that is getting caught in the filter, and occasionally there is a large amount of this. Some people may have to replace their filter every week normally.
As long as there's nothing that looks like sand and shells in the filter, everything should be ok.
I would venture a guess that @Tester101's suspicion about your floaties coming from your glass (unrinsed soap, dust, etc.) may be heading down the right path. I would try rinsing a glass, scrubbing it with a new sponge or clean washcloth, wiping it dry with a clean towel (not paper towel, as this may leave pieces of paper behind), and then refilling it. @NiallC's comment about asking for a water report from your municipal water supply is a pretty good alternative, but doesn't factor in anything that may come from the pipes between them and you. Most health departments will do water testing, so you could always fill a bottle and take it in.
Beyond that, if you really have to know what's in the water, your only real solution to absolutely know with 100% certainty (technically, unless you watch the health department do the tests, and can verify their results, you can't KNOW that they're right...but I'd probably just accept it) is to get testing kits for this, that, and the other thing (that was just the first page offering a wide range of kits that I found), that operate similar to what you might use for pool water. Depending on the sensitivity levels, you may pick up on things like trace amounts of chlorine...but of course this is going to get back into the more expensive end of things, as those kits aren't cheap (Grainger's seem to run around $40-$50 and up...it's about $20 for their cheapest, which is testing for copper, on that first page of results).
You could pick up a pool test kit, which will cover a couple of parameters (chlorine, calcium, hardness, etc.), but it's not going to cover everything and may not operate in the range you need to identify your floaties...it's designed primarily to help you maintain clear-looking pool water that you're not drinking, not crystal clear tap water.
So really:
Ask for report from municipal water supply
Take in sample to health department
If you're still paranoid and want to know what it is, buy test kits
If you just want to get rid of the floaties, filter, filter, filter, and make sure your dishes aren't the source
Best Answer
I think that you would be better served by a water conditioning company. When I moved into my new house 42 years ago I had a similar problem and I tried to correct the problem myself. When they failed I called "Culligan" who was the best company in my area at that time. A great gentleman came and tested my water. He recommended a solution that I thought was very expensive. I called a couple other companies but they were all reluctant to sell me anything. The guy from Culligan solved my problem. That man stated that my water condition was the worst that he had ever seen. 20 years I sold that house and the system still worked well with a minimum of service calls. I am glad that I called in a professional. my 2 cents