Without getting all complicated and stuffing you full of way more information than the average beginner to intermediate hobbyist needs I will attempt to steer you in the right direction as opposed to naming names or telling you exactly what I use and how. If you want to see what my test kit looks like try this link.  Test Kit Picture   I have another one that is just about as loaded.

    The basic concept of all tests are simply to give you a general idea of the state of the tank at a given time.  Unfortunately they are not meant to be the end all answer for troubles, nor are they an oracle that will give you all the information you need to deal with your tank.  To many "test tubers", as I call them, think they are the end all to anything that possibly could be in the equation for a good tank, or conversely a bad tank.  Simple B.S. is what I call that philosophy.

    Let me illustrate this a little for you.  Take the simple gas gauge in any automobile.  They generally have markings in quarters and possibly eighths right?  Do they tell you in gallons?  How about pints? or even better yet drops?  Not likely!  Here's why. It simply isn't necessary to have such readings for the intended application.  Who cares if their auto has 37,000 drops of gas? You just need to know about how much to within an accuracy sufficient to allow you fill up before you run out of gas. Nothing more nothing less.

    The same thing applies to testing your aquarium water.  For  the most part no hobbyist needs nor should spend their money on lab grade testing equipment.  In my opinion one doesn't need anything anywhere near that, and many of the available tests generally aren't needed at all.  Yet many a hobbyist or pet store worker will drill you asking for all sorts of water parameters and how they were measured even before they will begin to comment on anything.  Granted some are likely to help you pinpoint trouble and can give an overview of general tank conditions but just like your automobiles gas gauge can't tell you there is water in your gasoline, test kits can't possibly give you information that is beyond what they can do.  No matter what grade or level of accuracy your test kits are they should not be solely relied upon to answer anything more than what they were designed to answer.  Keeping this in mind when using them will put you miles ahead of the "test tubers".

    Consider all of them rough estimates of general conditions and things should make a lot more sense to you.

    Now that we have that covered, know that many are asking for these parameters for the same reason as you may want to know them.  To get a general view of the current conditions of the tank.  So what exactly are the general conditions and parameters of a tank that are of concern or may be tested you may ask.  Well, see below.

Size:  Tank size in both gallons/liters and physical dimensions.
Lighting:  Light type, wattage, spectrum, duration, dates of fresh bulbs, any changes in lighting, any sunlight hitting the tank (directly or indirectly).
Temperature:  Actual temp, Temperature swing in 12 hours and 24 hours, any spikes up or down.
Water changes:  Percentage of volume changed, duration between changes, brand of salt used, type of water used (tap from well, tap from city, tap parameters, RO, DI, RO/DI, filtered, etc.), conditions of mixed and premixed water, method used to deliver to tank.
Top off water:  Duration between top -off, type of water used, conditions of stored water, method used to add it to tank.
Water Movement and Aeration:   Amount of total flow through tank, through sump (if present). Type, brand, and size of pumps, placement of pumps, wave maker type (if present), air stones, skimmer (brand, size and model and pump that drives it information).  Direction of water flow, surface agitation, flow in all areas of tank.
Feeding: What do you feed, when do you feed, how much do you feed, method of feeding, water pumps on or off during feeding.
Livestock:  Virtually anything that you can identify should be on this list including it size, general condition, and age.
Other hardware:  Any other hardware used such as, but not limited to, Skimmer, Bio-balls or the like, Power filters, Ground probe(s), heaters, GFCI's, canopy, fans, etc.
Water Quality/Properties:   Ammonium, Nitrite, Nitrate, Ph., Calcium, Alkalinity, etc. this list could go on and on until there simply was nothing else to test for, but these are the most helpful.  Of these the main 4 are the staple of what I believe you  should test for when you test.  They are Nitrate, Ph., Calcium, and Alkalinity.  Unless you are cycling your tank, have added massive amounts of anything to it, or have reason to suspect something is amiss you generally do not need to test for the other two.
Specific Gravity:  Should equal as close to 1.026 (actual) as you can maintain. Please don't torture yourself with this.  Know your tanks temperature and just "guesstamate" it within .001 and all should be well.  If you are using a standard floating hydrometer, such as those you buy in LFS, it should read about 1.026 at 75 F and about 1.025 at 80 F to equal normal seawater salinity.
    A little caution, or a lot as the case may be, should be used to confirm that your hydrometer is calibrated where you think it is.  Also use proper cleaning and storage methods.  That means wash it out with RO, RO/DI, or Distilled water after use and keep it in a sealed environment.  Sealed does not mean under water in your sump!
    Or if you prefer to be a scientist with it get a good hydrometer in the "bobber" form and keep it perfectly in line with 1.026 at 60 degrees.  This means that if your hydrometer is calibrated at 60 degrees and you run your tank at 75-85 degrees (as most are) your reading should be about 1.024 and 1.023 with 1.0235 at 80 degrees being the half way point.  Good Luck!  When you give up, or tire of tweaking everyday or every other day, read my comments above about the automobiles gas gauge.
    Another option is to use a refractometer.  These are usually automatically temperature corrected, but make sure you confirm that.  Also you will need to make sure it is calibrated correctly and not abused.
(Thanks to Boomer for his help with the Specific Gravity section.)
Ammonia = 0
Nitrite = 0
Nitrate = as low as you can maintain it without putting you and/or your tank through hell getting it lower.
Ph. = Generally accepted to be held to swings no higher than 8.6 nor lower than 8.1 with the ideal being 8.3-8.4.  Note that daily (even hourly) swings are to be expected so don't get all bent out of shape and toss in a bunch of additives unless your sure you need them.
Calcium = Generally accepted to be between 400ppm and 450ppm, trying to hold near 425ppm seems reasonable.  A little low, say in the 380 range is often acceptable, but the higher you go past 450 the most issues you can run into.
Alkalinity = Not below 2.5meq/l nor above 5meq/l, but try to stay as close to 3.5meq/l (10dKH) as you can.  A little variance in alkalinity should not cause you to get in a panic and start dosing.  A little low is generally worse than a little high however.
Phosphate = If you want to test Phosphate I suggest you test the levels in the water you add to your tank and not the tank itself.  Maybe even disolve some of your tank foods in water then test that water for phosphate... use some good math and figure out the influx into your tank.  This could even help you choose better foods for your tank.