If you do not already have a web mail account:
You can create a new Gmail account using your cell phone.
You can create a new Yahoo mail account using a computer (or maybe a PDA). After you create your account, your can check mail using your cell phone.
You can continue using the above Email even if you switch cell phone providers. Your phone browser can probably display Gmail and Yahoo webmail pages because those servers send simple pages for small screens. Try comparing the links viewed on a computer vs. a cell phone.
Webmail and SMS texting are completely different.
Cost:
SMS costs per message unless your plan includes unlimited SMS messaging
Webmail costs per kilobyte unless your plan includes unlimited web browsing (internet access)
How to access:
SMS features very short text messages, and is probably available on your phone as messaging.
Webmail: Text length is nearly unlimted. You navigate to a web page to do web mail (your cell phone company's web mail may be an exception).