Was my water softener hooked up correctly?
I can't tell from the picture which is the inlet and outlet, so I will make the assumption it's not hooked up backwards.. 

However, there are a few things I see that I don't like:


The hole cut in the drywall is pretty unprofessional
There is a shut-off valve in the wall - which means you need an access panel. If the installer added that there, he's incompetent. If it was there before, he probably should have highly suggested to you he moved it out -- if he also did the plumbing (if he cut the hole as you said, then I suspect so) then this would have not been much extra work. 
The overflow on the brine tank doesn't go anywhere (the black barbed elbow coming out the side). I have seen at least two basements who had a couple inches of water (ruining carpet, etc) due to a failed $10 shut-off float valve in the brine tank. 

Yes you can fix the drywall, but don't cover the valve (or leave an access panel). The valve has the potential to leak, and doing that in the wall is bad. You may also need to shut it off at some point (eg, to disconnect the softener). The best option would be to replumb the valve so it's not in the wall.
Sorry I don't know enough details about this particular model or brand to answer specific questions about your model. See if you can find the manual, especially the installer's manual. A quick google search turns up some promising results. 
You can get hardness test strips to check the hardness before and after. There's actually some fairly complex chemistry involved in "testing water" (which, frankly, most water treatment companies don't fully understand), and I could go on in a lot of detail about fully testing .. but I'm not even sure that's what you're asking about so I won't get into it.

To set up your softener correctly, you basically need to know the hardness of the water. It looks like your softener probably has a flow meter in it so it automatically recharges based on usage; some of the more basic models recharge based on a schedule and so you need to have a rough idea of your average daily usage as well to know how often to recharge them. 
Softeners are pretty easy to maintain. Make sure there's salt in the brine tank. Every 10 years, you need to rebed the media in the filter. Make sure nothing is leaking, and that it's recharging (usually it does this during the night, anywhere from every day to once a week, though could be less if you use very little water).