You don't need wires.
Of course that's not true; your gadgets still need a cable, but it's closer to true than it has ever been. If you're willing to put in a little time getting set up, and possibly invest a little in hardware and services, you can get your tech organised in such a way that wires and even your physical location are all but irrelevant to daily use of your computers and gadgets. Wireless operation became important to me when all of my computing devices became mobile. I really want to have full and complete functionality regardless of where I am at any given time; access to all my stuff, my files, my music library and movies. It shouldn't matter if I'm in the street on my iPhone or the sofa with my MacBook. This won't necessarily all be suitable for everyone, but some of it will for sure.
I'm going to talk about Apple kit, because that's what I use. There are comparable alternatives for Microsoft, Linux and Android based setups. There are options that work perfectly well across platforms. I'm going to cover what I know but there's more information out there.
This is the cable you can't do without. Although there are some 'wireless' chargers out there for phones and such like these still need to touch (or at least be within millimetres of) your phone. They're cool, they're convenient, but they don't offer the real benefits of a wireless solution. Right now, power is a cable-driven deal, but most of us probably charge things in the same physical place day by day, or night by night as the case may be. Wireless charging is not on the whole not radically beneficial when compared with just plugging in.
I'm really mentioning this for completeness. You're probably already using Wi-Fi at home and elsewhere. I think it's worth mentioning that most phones now are capable of operating as personal Wi-Fi hotspots; a feature I use almost daily to connect my iPad to the net while I'm not at home. Modern Wi-Fi is fast and reliable, I don't think I need to sell you on it's use or it's benefits.
Storage is the other difficult area. If you are dealing with high data volumes and need high speed date transfers then you are probably stuck with external drives connected by Thunderbolt or USB 3. If you don't have this issue (and the average computer/smartphone user probably doesn't) then you can start to consider cloud storage as a solution for the bulk of your file handling.
There are two kinds of cloud storage: an online service or self-hosted. The difference between the two is who owns the hardware (and who has to troubleshoot it if something goes wrong). You may still need to keep sensitive files on your computer's local drive but services such as iCloud, Dropbox, Google Drive, and many others offer degrees of file syncing and remote, centralised, easily accessible storage for your stuff. If you're interested in hosting your own cloud storage my recommendation is a Transporter.
With cloud based storage you can treat your external storage similar to as though it were physically connected to all of your devices. Without a data connection however you're either reliant on synced local copies or out in the cold. For most of us, access to mobile data or Wi-Fi is probably very reliable.
Immediate file transfers are also pretty easy on Macs and iOS devices. AirDrop, previously a slightly awkward Mac to Mac wireless file transfer now works between all Apple devices that have newer Wi-Fi and Bluetooth support. You can quickly transfer files, web pages, etc with a few taps. No more passing around a Flash Drive. This is not only convenient but also safer.
Another place where you might be able to eliminate cables is your desk. While you may need a cable to connect your monitor to your computer there is no longer any particular reason to use wired input devices. So long as you're able to spend above the most basic budget models a wireless keyboard and mouse will not suffer the connection and battery issues that they might have a few years ago. I would find it hard to go back to dragging a mouse cable around on my desk; for the rare inconvenience of swapping out some rechargeable batteries I have more desk space and less clutter.
Your Music and Videos
Actually, your media in general. A little over a year ago I started using iTunes Match, which is half music-file amnesty service and half music streaming service. It streams you your own music library and Google and Amazon have comparable services available. There are also music streaming services such as Spotify and Rdio that stream music in a format more comparable to radio. Either of these types of services can offer you wireless access to music depending on your preference. iTunes Match also allows you to stream movie and TV content that was bought from iTunes without pre-downloading it to a device (beware mobile data use). Match works well for me, but isn't going to be suitable for everyone.
You could also host media files on cloud storage, so long as the storage and data requirements are not prohibitive in your circumstances. This is getting easier as connection speeds are increasing over time and cloud storage is getting cheaper. Another streaming concern is device to device on your local network. I use an Apple TV and AirPlay enabled speakers to stream media around my home. With AirPlay there is more limited file format support than on some alternatives.
The additional benefit to me for using iTunes Match is that I no longer have any need whatsoever to sync my iPhone and iPad to my Mac. Everything happens via iCloud and iTunes Match now.
Apple very recently launched iCloud photos, which is centralised, synced storage for your photo library that is built in to the iOS and OS X photos apps. You could achieve this with cloud storage options; Dropbox and others have some support for this built-in to their apps. The advantage of Photos is that the syncing system preserves editing history non-destructively (you can make an edit in photos on your iPhone and tweak or discard that edit on your Mac later) and it's well integrated into the devices sharing functionality. It's also smart enough not to use up all the storage on your device, even if your photo collection is significantly larger than your device storage.
iCloud photos has been great for me so far, though the photos apps on both iOS and OS X need to mature. If you're a professional or serious hobbyist photographer it may simply not be powerful enough for you.
This isn't so much about wirelessness as it is about disclessness. With modern internet speeds and digital distribution platforms like the App Store and Steam few of us are likely to have purchased software that comes in a physical box for a while. The exception may be console gamers but even games consoles are coming to embrace digital distribution now. This is equally applicable to purchasing music, movies, and in some cases (in a sense) even physical objects can be downloaded with home 3D printing now a reality (if an expensive one).
Even backing up your data is now a simple task to complete wirelessly. With Time Machine on the Mac and a Time Capsule you can automatically and wirelessly back up regularly while on your home network. I also maintain a remote backup courtesy of Backblaze which is a continuous sync that will store any quantity of data for a month. They'll even ship you your files on an HDD if a calamity happens. Having numerous copies of your data on different devices and in different locations (sometimes called a 3-2-1 backup system) is the best way to keep your stuff safe. There are other similar services out there, so you can look around until you find the one that you like.