Make sure your port names and numbers match exactly between the old and new stacks. This means that for example, GigabitEthernet1/0/1 is on the top most switch and is actually named GigabitEthernet1/0/1 on both switches. Since this is 3750 to 3850, that shouldn't be a problem. You just want to make sure the top switch is 1, the second switch is 2, etc. on both stacks. You can use the show switch detail to make sure and the switch number (I think) command to make any changes.
Where you might run into issues is if you use a different SFP port for your fiber uplink trunks. Make sure those are in the same ports to make porting the config easier.
Set up a TFTP server on a PC and use the copy run tftp command to copy the config over to the PC.
Set up an IP address on the 3850 stack (and gateway if the PC will be on a different network) and use the copy command (i.e. copy tftp newconfig) to copy it over to the new stack.
I believe 3850s also have a USB port so you can play with putting the config on a thumb drive and copy from the USB port (disk0: I think) to the flash in the switch.
Do a copy newconfig start command to copy the new config to the startup-config which is what the switch will use when reloaded. newconfig is whatever you named the file when you copied it over. start is short for "startup-config"
Do a show start command to make sure the copy took.
Do a reload command to reload the stack and monitor the reload for errors. There might be commands in the old config that have been changed or deprecated. Cisco usually does a good job of handling these assuming the 3850 is running a newer version of IOS than the 3750.
Make sure you are connected to the switch via a console cable when you do all this. The same PC that the console cable is connected to can also be your TFTP server connected directly to the switch to make it easier.
You may also have license file issues to deal with. At the very least, make sure that if your 3750s were running IP Base, the 3850s are at also running IP Base or IP Services and not LAN Base. The 3750s will probably be running the IOS 12 train which I do not think uses the new license PAK scheme. The 3850s are probably running IOS 15 train which may need license codes to enable certain features.