When you are asked to provide answers to a security question ("What is your mother's maiden name?"), you don't have to answer truthfully. Your account will actually be more secure if you make up an answer. Just make sure you remember the answer you've made up!
LastPass is a password management service, offered through the web. It generates randomized passwords for sites you need to log into and stores them, logging you into sites only when you provide your master password, which you generate and which is never stored.
KeePassX is a password management application that you install on your own computer. Instead of storing your saved passwords on a company's servers, you set up your own database to store them on your own computer.
A Yubikey is a special kind of USB drive that lets you add a second layer of security to a log in. In addition to putting in your username and password to access a site, you also have to insert your Yubikey and press a button on it. This means someone hoping to access your account would have to steal both your information and your USB drive.
Much has been written about what makes a strong password, and a lot of different techniques for strong passwords have been offered online. A few articles on the topic are linked below. If you're creating your own passwords, the most important things to remember are:
How to Make a Strong Master Password
How To Create Super-Strong Passwords To Protect Yourself From The 'Heartbleed' Bug