The secret to doing all of this is to use Dropbox. Dropbox is a popular cloud-based service; place files in the Dropbox folder on your computer and they are automatically copied (synchronised) to a private, corresponding folder on the internet. Now install the Dropbox client on another computer; as if by magic your files start to appear there. Add a third computer and they appear there, too. At the most basic level, one of the main purposes of a network is to provide a shared area in which items can be stored and Dropbox constitutes that shared area.
Assuming you are starting from scratch, the first thing to do is to create a Dropbox account, which you can do by going to the Dropbox website. Then, install the Dropbox client on all your devices - Windows PCs, Macs, iPads, Smartphones etc. Decide which items you want to share and copy them into the Dropbox folder on your computer and shortly thereafter they will appear on all the other devices. On the PCs and Macs, make sure that LAN sync is turned on, as this will improve performance. To do so, choose Preferences and go to the General tab of the Dropbox clients.
What are the possible pitfalls? Well, keep in mind that the files in the Dropbox folder are 'live'. That is, if someone deletes a file on one machine then it will shortly vanish from all the others. For this reason, be sure to keep separate copies of important items outside of Dropbox. The other main consideration is storage space: this is not a solution for people who want to share movies and stuff. With the standard free Dropbox account there is just 2GB of space (but if you haven't got an account yet send an email here with the word 'Dropbox' as the subject and we'll send you a link to start off with 2.5GB instead). Once you are up and running you can obtain additional free space by recommending your contacts and by linking your Twitter and Facebook accounts, up to a maximum of 16GB. Thereafter you would need to pay for space, but at that point you might be better off considering a proper NAS system.