Maple City Woodsman:
The .32 Double Action 4th Model was manufactured between 1883 & 1909 in the serial range 43406 to 239600. if your gun has a serial number 377xx then it is a 3rd model manufactured in 1882 or 1883 in the serial range 22173 to 43405.
mjdio:
You have a .38 Military & Police Model of 1905 4th Change manufactured between 1915 & September 1919. Serial range for those years was 241704 to 316648. Your gun lacks the modern hammer block safety. If dropped it could fire, so leave the chamber under the hammer empty. it also pre-dates heat treatment of cylinders, so stick to standard pressure lead loads only, no jacketed, semi jacketed, PlusP or +P+ ammo. Use of these may result in a bulged or cracked cylinder.
Value is probably in the range $250 to $350. If it has factory adjustable sights or factory pearl or ivory grips the value may go up a couple of hundred dollars if you can find a collector wanting to fill a niche. Several million M&Ps have been manufactured over the last 115 years, so they aren't particularly scarce or valuable.