There are different shortcuts for accents depending on the operating system that you usually use (Windows or not-Windows), the layout your keyboard uses habitually (e.g. US, UK English, or another language), and the size of your keyboard (laptop/ full size, & etc).

I've added a French keyboard permanently to my Windows language options. I can't make any of the shortcut options work on my small keyboard, but adding a second keyboard language is simple and gives me all the accents that I might need. Seems to work for e-mails, documents, internet and everything.

Another option which might suit you uses the US International Keyboard as a base: Find and add the US International keyboard. This has dead-key functionality on accent characters such as `'^~ etc. So you type 'e which gives you é and so forth. It's quick, intuitive, doesn't require memorisation of codes or particular keys.

The only problem is the way it is nested away.
User Settings > Time and Language > Region and Language, install English United States, click on this and select Options, then add United-States International Qwerty as the input method, then remove the standard US keyboard (if you're not using it).

Switch between keyboards any time with Windows key + Space.

Hope that's useful and not my usual bag of hot air !