I doubt that any major changes would be made in a minor release change from 3.8.2 - 3.8.11.
If SQLite uses Semantic Versioning like many projects do, the only time major changes that would affect compatibility are made is between major versions (eg. 3.x.x -> 4.x.x in this case). Thus, you should be okay to upgrade it, but of course you should read over the changelogs for confirmation.
As for the proper way to do it, if you keep your system up to date, then you will have the most up to date version of SQLite3 available to your Ubuntu release available in the official repositories. You can make sure of this by running
sudo apt-get update && sudo apt-get upgrade && sudo apt-get dist-upgrade.
As it stands currently, the newest version of SQLite3 available to 14.04 LTS is 3.8.2, as per this page. The latest version available through official repositories period is the 18.104.22.168 release, available to the 15.10 release.
Thus, if you want to use 3.11.x on your system, you will have to do it manually from sources provided by SQlite.