I don't know if this is really an answer, but I've had a Coleman 6 person spa in service for about 25 years. Yes, 25 years! I have never had the same problems you are experiencing. We learned early on that any clothing (bathing suits etc) must be very clean and rinsed extremely well before going into the spa. Clothes cleaning detergents left in the fabrics release into the pool water and cause foam quickly. The clarifier is good at flocking these soaps, but they do tend to gum up the works quickly. We never lost a lot of pressure however, just had to clean the filter. We also made a habit of completely changing the water at least twice a year. After draining as much as we could via the drain hose, we used a wet vac to get as much out of any low spots. We then put in just enough water to cover the intake close to the bottom and jogged the pump to cycle clean water through the system and get the dirty water out. Again, drain and vacuum and repeat this process until the water looks clear. We do this with the filter out. This will clean out a lot of crap. About once a year just before we do a drain and complete cleaning, we treat the tub with anti-scale cleanser. This cleans the internal pipes of built up scale and solids. In your case, you may want to do this anti-scale treatment for a few days with the circulator on low and the filter out to purge the internal system. Then go ahead and change the water.
Another test you can do is to see how well the pump pushes water. When you have the small amount in the bottom, just above the intake, jets exposed, turn the jets to full, air off and see if the pump will shoot the water out of the jets with extreme force. Mine will shot water 20 feet away!!!! lol. If you don't have really good pressure, you may have a warn pump impeller. They can be damaged over time by chlorine based sanitizer, I never use chlorine, just bromine based shock etc.
The other item to check is if you have the right filter for your unit. Filters come in so many styles and densities. You may need to find one that allows better flow.
Assuming you find the root cause of your problem, good housekeeping, keeping as much contaminates as possible out of your tub will help. Very clean bathing suits or none at all,(yahoo) no body cosmetics such as body oils, moisturizers, perfumes etc. Keep chemical treatments to a minimum. Do a quick check weekly for PH, Total alkalinity, and bromine level. Adjust them sparingly so not to overdo any chemicals. Once you get everything under control, maintaining it is very easy and quick. Good Luck
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
No, a sediment filter will not remove calcium -- the only thing that will is a water softener. These use a media that only needs to be replaced every 10 years, and a brine tank with salt, which you have to refill periodically depending on water usage and hardness levels: anywhere from a couple weeks to a couple months. There's really no other regular maintenance that normally needs to be done on a softener.
Softeners substitute hardness (calcium carbonate) as well as some heavy metals, lead, manganese and low levels of iron with sodium ions (salt). The drawback to this is that increases the sodium levels in the water, which can be a concern for people on low-sodium diets. The amount of sodium added is entirely dependent on the amount of hardness removed.
For this reason, people often run a bypass to the kitchen cold water tap so it gets hard water, as well as outside taps used for watering the lawn and garden.
Before you purchase, you should test your water for hardness. The easiest and cheapest way is to buy a hardness test strip kit: you dip it in the water, match the colors and it tells you the hardness level. This will help you sizing the softener to buy. It's also good to periodically check so you can ensure the softener is set correctly. You're on city water, but depending on where they get the water from the hardness could change seasonally.