You want the green valve for normal water shutoff purposes
If you're just seeking to shut off the household (domestic) water to work on the plumbing, you want the green valve, since that will shut off your plumbing without shutting off your fire sprinklers.
The two blue valves are part of your fire sprinkler's backflow preventer
From the arrangement of parts there, we can see that you have a classical fire sprinkler riser/drain installed, with a waterflow switch (Potter is one of the major manufacturers of such parts) that sets off some sort of local alarm (such as your smoke alarms, or even a local bell/horn) and/or interfaces to an alarm panel that can call whoever's monitoring your alarms if the sprinklers ever start flowing water, presumably because a fire caused one of the heads to open.
The white pipe is a drain pipe from the test drain on the system; opening the valve connected to it (located below the waterflow switch) will set the waterflow switch off, and should be done as part of the normal annual checks of the system. Of course, if you have an alarm panel and monitoring company, they'll need to be informed of what you're doing beforehand so they don't dispatch the fire department to a nothingburger!
Going back to the blue valves, though, those are the shutoff valves for the backflow preventer your water utility/local jurisdiction requires to keep sprinkler water (which can get very stagnant, unlike your normal house water, which is used on a regular basis and thus doesn't need any protection beyond the normal measures of air gaps at fixtures and vacuum breakers on hose bibbs and irrigation systems) from backing up into the domestic water system in case of a water main break or water pressure shortage. These need to stay open at all times, as the sprinklers won't have any water to do their job when they're closed. In fact, they shouldn't be left unsupervised as they are now; instead, they either need to be chained and locked open with a good quality lock, or have tamper switches hooked up to them so that they can generate a supervisory indication to your alarm panel when shut, provided you have an alarm panel that's capable of that.
The reason why they're there, and the primary reason why they need to be shut, is because the backflow preventer itself must be tested on a regular basis, and the test procedure requires the use of those valves for short periods of time. (This is what you've been witnessing the plumber doing -- the gadgets they use for this purpose are merely a set of pressure gauges that let them "see" what is going on within the backflow preventer, since it runs entirely on water pressures.) They can also be used to shut down the sprinkler system so that a fire protection contractor can drain it to do work on it, or in case a sprinkler head gets whacked or activated by a fire that's since been extinguished and overhauled, and thus needs to be replaced with a fresh head.
The red valve is a master shutoff, and should also be locked open or supervised
Going upstream, past the water meter on the domestic side, we then come to the red valve. This is in the incoming water main before it splits into domestic and fire branches, and thus shuts off both your domestic plumbing and your fire sprinklers when closed. As a result, unlike the blue valves, you'd at least stand a chance of noticing if someone shut it off out from under you, but there's still a cogent argument to be made for locking it open or electrically supervising it in the same fashion as the blue valves.
Furthermore, since the meter is in the branch feeding your household plumbing, your fire sprinkler supply is unmetered. Given that it'll only ever flow a non-trivial quantity of water in case of a leak (damaged sprinkler head) or a live working fire in your house, most water purveyors don't care to meter fire sprinklers, instead treating it like the fire department using water from a hydrant to fight a fire, and at most charging a fixed fee to you for having a sprinkler system.
Other notes for a fire sprinkler owner
Since you have a fire sprinkler system, there are a few other things you'll need to keep in mind. First, whole-house water softeners or filters should only be cut into the domestic branch of the system; fire sprinklers will run on pretty much whatever water the utility supplies, and softeners and filters can have very negative impacts on the ability of the system to supply enough water to successfully fight a fire until the FD can get there and finish the job.
Second, there should be a cabinet not far from here that has a funny looking wrench/socket and a set of sprinkler heads in it. The wrench is a sprinkler wrench that's used to remove and install heads, and the heads in that cabinet themselves are spares. This is important since a sprinkler head is a one-shot device akin to the fuse in your Christmas lights -- once it goes, you can't put it back together again, and must replace it instead. There should also be a set of plans in that cabinet -- these will be the hydraulic plans your sprinkler system has been custom-installed to, and are very important to provide to your fire sprinkler contractor if you are having major remodeling done on your house so they can recalculate the plans and thus ensure that the new work is as well-protected as the old work is.
Finally, whenever you're painting the walls and/or ceilings (whichever surface your heads are mounted on), you'll want to mask off the sprinkler heads with tissue paper bags, as paint splattering onto a sprinkler head essentially ruins it in the eyes of the fire code -- there's no telling what impact the paint could have on how the head behaves in a fire. (The tissue paper will cleanly burn away if there's an actual fire, which is why it's used in this instance -- it's a trick taken straight from the fire sprinkler code itself, where the bags are used to protect sprinkler heads in paint spraying booths from the inevitable paint overspray found there. See NFPA 13 paragraph 26.4.2.2.1 for details.)
