As for steps when trouble-shooting something like a storage device, I usually check the following:
1. Is the USB cable broken?
Try a different cable
2. Is the USB port on the computer broken?
Try plugging in a device that you know works and verify that the port you are using isn't broken. If it is, try to find a port that isn't broken!
3. Is the device broken?
The device itself might seem OK, but if its USB port is broken/damaged, that wouldn't help with connecting to it!
Try plugging it in to another machine (your old system, a friend/relatives) to verify that the device is not broken/damaged/malfunctioning.
4. Does the device use any unusual protocols (e.g. MTP) or exotic file-systems?
This usually involves a little duckduckgo-fu. Try to find out a little about the device and whether you are able to use it with Linux.
Ensure you have the appropriate software/tools installed to be able to support these protocols/file-systems
After all of the above:
5. Is the device recognised by the kernel?
If you have verified that the device is working and that you have appropriate tools/software installed to be able to connect your system to the device, the kernel should be able to recognise the device when it is plugged in.
To verify, you can use:
to see the last few messages from the kernel.
With any luck you should see some identifying information about your device.
In the case of mtp devices, you could also try using:
If the device
WAS recognised, but you are still unable to connect via your file-manager, it may be some quirk/bug with your file-manager.
In which case - as a last ditch solution, you could try manually mounting the device via the command-line. I won't go into how to manually mount a device here because this post is already well into TLDR; territory.
If the device
was NOT recognised, then the chances are that the device requires a specific driver.
Some distros use slightly older versions of the Linux kernel. It could be that there is a driver for it in a newer version of the Linux kernel. Again, you'd have to do some keyboard kung-fu with your favourite search engine to research this.
If a driver is available in a newer version of the kernel, you can either install a newer version from your distros repos (if they have one) - or you could build and install it from source yourself.
However, if the device is extremely new, or very uncommon/obscure - it is possible that there are no drivers for it at all. But like I said previously,
most storage devices use standard, generic protocols and file-systems and
SHOULD be accessible/usable on Linux.