Chevy Truck

Has the transmission ever been serviced? Check the fluid. If it's brown, it definitely needs flushing. If the problem has been occurring for awhile now, you may need to replace or rebuild the transmission. The speedometer issue tells me there is either a problem with the computer or with the vehicle speed sensor.

You could try testing replacing the neutral safety switch as that tells the trans controls what gear it is in? I've heard of other vehicles fixing issues by replacing this item. Speed sensor also. Try to find out if there are any other sensors related to trans function.

How many miles on the new/replacement transmission? Since the speedometer is electronically-controlled, the problem with it is definitely electrical. Make sure the connectors on the back of the dashboard are firmly seated. Do the "wiggle test". Get a multimeter, set it to read resistance and wiggle the wires and connectors while watching the meter closely. You should see no fluctuation in the readings.
The slipping problem could be quite a few different things. Check the fluid for color and smell. If it is brownish or smells burnt, get it flushed. If you have an Aamco near you, they do free diagnostics on transmissions.

Find somewhere that will do a free diagnostic. Shifting back and forth between 2nd and 3rd tells me either speed sensor or vacuum modulator, if so equipped.

My pleasure