I use Oulook Express for my e-mail at home where my ISP is Charter. When traveling where there is a different ISP, I can not use my regular Outlook Express, I must go to the Charter site and use their e-mail which is not convenient. Is there another e-mail system which would not give me this trouble? What is it?

If you have another free email service, you could forward your email from your outlook email address to your free email account

Here is a link with some info
http://ittraining.lse.ac.uk/Documentation/OnlineGuides/Forward-your-email.htm

Yes and no
If you are running a mail client (outlook express ) locally (on your computer) configured to get mail via POP or IMAP or send mail using SMTP you would be able to connect to the mail servers once you have established a internet connection. It should work like accessing any other server on the web.
The reality is that many corporate and community networks block access to specific ports to protect their network and mail servers from being blacklisted as spammers. So your Outlook Express may not be able to send mail. This is unfortunately becoming more and more prevalent. Gmail can be configured to run in Outlook express is free and uses a non standard port to send mail so it is less likely to be blocked on networks blocking SMTP prot 25 traffic.