After initial startup voltmeter says my battery is at 10 volts, the battery saver mode comes on and it says to check battery charging system. Then after turning it off and starting it up again immediately it reads at roughly 13.8 volts and climbs to roughly 14.5 volts. If I don't restart my car immediately my car will run but everything will slowly turn off in my car and my car will turn off. But if I restart my car it will stay roughly at 14.5 for the entirety of my trip. As if I us my car again within 10 hours of the initial use that same day I don't have to restart it. My vehicle is a Chevy Equinox '07, I have replaced the battery, starter, and the alternator all within a year ago.
You have a parasitic drain on your car, the battery voltage should not drop that low(unless it was sitting cold). You need to find the cause ,try using parking inside a garage(remeasure voltage next day) or use a clamp meter on amps and start pulling fuses from the fuse box to see the current go down , and isolate the problem (use electrical diagram from service manual for your car) .Note when you first unlock your car , the fuel pump/proximity circuits will start , you need to wait for that to settle down , then measure current whilst the car is off.
See https://www.youtube.com/watch?v=KF1gijj03_0 for what i mean.
Consider the proposition of a faulty wiring loom/device , if there is a short/open circuit (intermittent fault),pulling the voltage down , removing fuses may isolate the circuit if there is a short .All that we've so far established is, that the circuit operation normalizes after a restart. During that restart period connections could be tightened due to heat , or even have better connections due to the vibration jostling the connector/loom.I would starting checking connections to the big 3 and main battery connectors.
Playing off your point , the battery is charged using the regulator, via modulation by the ecu output , wiring/plugs could be damaged there as well , check voltage on alternator regulator plug/wiring loom (be careful of moving parts and belts with your hands/limbs/DMM leads).