I have their modem but use my wifi router--linksys 10378.
Are you sure that is the correct Linksys model number? A quick internet search did not turn up anything for it. I wanted to see if I could find a manual to see what potential functions it might offer regarding hooking up an external drive.
In general, there are two basic ways to "share" an external hard drive on a network: 1) leave it attached to a computer that is always turned on and then share that drive using the Windows (or whatever OS you are using...you did not specifically say, so I am assuming Windows) network sharing service/function; or 2) connect the external drive to some network device that allows for the connection of external USB drives to be shared to the network, where that network device can be a router, an NAS, or a dedicated device for sharing an external USB drive. Technically, this is a third option in the form of external USB drives that also have a built-in WiFi function, but those types of drives typically are not "true" network drives (i.e. they don't connect to the network to be shared), but rather are designed to have the computer, phone, or tablet connect directly to the drive using WiFi...thus only allowing one computer, phone, or tablet to connect to the drive at a time. And, of course, this last option would require getting a new drive as that option is only built into drives (to my knowledge), so if your existing drive does not have that ability, then this last option is not really an option...unless you are willing to buy a new drive.
If you want to try #1, then we can likely provide instructions. But, keep in mind that the computer that the drive is physically connected to will need to be turned on for other devices to be able to connect to the drive. And I will also warn that Windows sharing can be a bit "temperamental" in my experience. If you want to try this route, then you should let us know which versions of Windows (or other OSs you are using) on the computer you will leave the drive connected to as well as any computers, phones, or tablets you want to then connect to the drive over the network.
For #2, as noted above, we will need accurate model number information of your router to determine if it is capable of doing what you want. If it is not or you want to go another route (no pun intended), then you are likely looking at buying some additional hardware, whether a new router, an NAS, or otherwise. And I will tend to note that is correct that you would likely be better served just getting a NAS...and then you could use the external hard drive as a backup drive for the NAS and/or your computers...or for some other purpose (although I encourage the backup idea...unless you are already backing things up to another drive or media).
You also mention wanting iDevices to be able to connect to the drive, which I am assuming you mean iPhones, iPads, and/or iPod Touches. If so, you will need to realize that an iOS device will almost definitely not be able to do this by default (i.e. with built in Apple apps). You will need third party app(s) in order to do this. For videos (assuming they are not copy protected...i.e. stuff bought from the iTunes Store), you can use the free VLC Player. Personally, for general file access from iOS devices to my NAS, I use an app called FileBrowser. It can handle most file formats (i.e. pictures, videos, Excel, PDF, etc). And it should be able to connect to both a computer sharing files using Windows file sharing (never verified that myself) as well as a NAS (can definitely confirm it will do that).
About the only slight warning to offer is that some NAS device might using proprietary connections (i.e. you have to use their dedicated software to connect to the drive). I believe most have moved away from that, but some might still use such connections. If you go with a NAS, you will want one that using standard network protocols. For Windows, this will be SMB/CIFS. Macs can also connect using this protocol, but they can also using AFP (an Apple protocol). I believe Linux should also be able to connect using SMB/CIFS. Of course, there is also HTTP (i.e. web page protocol) or FTP.