Hotmail and Yahoo Mail are fine, but Google has a feature in their webmail that makes it much better to use. You can tag your email with labels, and then dump the emails all into one big folder. Don’t worry, because the search is fantastic. I can always find messages that I have saved.
And you can save a lot of them. Google gives you many megabytes to store your email messages. I even have Google go get my school email so I can access from anywhere.