If your Ubuntu computer has wireless, I highly recommend using FTP over wireless. It is convenient (no wires!), offers good transfer speed and the entire /sdcard contents are available for read and write.

If your Ubuntu computer does not have wireless, then your next option is to use a micro-USB-to-USB cable. The choices here are PTP and MTP. You can enable and switch between the two after you connect Moto G to your computer using a USB cable. If you enable PTP, then Moto G is automatically mounted as a partition in Nautilus. However, you will only be able to see the /sdcard/DCIM and /sdcard/Pictures directories. If you are transferring photos, then this option offers good transfer speed.

If your Ubuntu computer does not have wireless and you still want to read and write to the contents of /sdcard then the option left to you is MTP. Note that with MTP, Moto G takes a really long time to mount and the transfer speeds are really really low. It can take ages to transfer even a small file. You have been warned! 🙂

There are many solutions offered online to mount the contents of /sdcard using MTP.This solution is the only one that worked for me:

• Install the MTP packages:

$ sudo apt-get install mtp-tools mtpfs

• Connect Moto G using a USB cable to your computer. Make sure MTP is selected, and not PTP. 
• Find out the vendor ID and product ID of Moto G using mtp-detect. For my smartphone I got:

$ sudo mtp-detect
Unable to open ~/.mtpz-data for reading, MTPZ disabled.libmtp version: 1.1.3

Listing raw device(s)
Device 0 (VID=22b8 and PID=2e82) is UNKNOWN.
Please report this VID/PID and the device model to the libmtp development team
   Found 1 device(s):
   22b8:2e82 @ bus 1, dev 12
Attempting to connect device(s)
Android device detected, assigning default bug flags

You need to press Ctrl+C to stop the command. For the Moto G, you can see that the vendor ID is 22b8 and product ID is 2e82.

• Open a new file /etc/udev/rules.d/51-android.rules using sudo and add this line:

SUBSYSTEM=="usb", ATTR{idVendor}=="22b8", ATTR{idProduct}=="2e82", MODE="0666"

• Restart the USB service and create a directory to mount the Moto G:

• Open the /etc/fuse.conf file as sudo and uncomment the line foruser_allow_other 
• Restart your computer. Connect back the Moto G to the computer.
• You can now mount the /sdcard of your Moto G using this command:

$ mtpfs -o allow_other /media/motog/

Note that the mounting operation is slow and might take about a minute.

• You can find all the directories and files in /sdcard of Moto G in /media/motog. You can read and write to these directories. 
• To unmount use this command:

$ fusermount -u /media/motog/

That is it! You may want to create aliases for the mount and unmount command to make it easy to use 🙂

Note: You will not get the USB mass storage option in Moto G since that can be provided for external storage, not for partitions from which the Linux kernel is currently running. And in any case, you will need root access on the phone to touch those directories.

Tried with: Moto G and Ubuntu 12.04