In the first case, you simply go to your Inbox mail in order to check it. Use your login and password in order to do this, then either keep track of a tab with a page of your mail, either periodically refresh the page. You can also click on the link "Inbox" from a page of your mail.
If your mail supports installing any kind of add-ons, install them in order to track new messages, not performing unnecessary actions. In this case, in your browser or in windows sidebar will be integrated label indicating the presence of new messages with which you will be able to stay knowledgeable about them and instantly go to the page of incoming emails.
Use Microsoft Outlook or another email client to collect, send and save letters. The program will update the folders of incoming and outgoing emails after a certain period of time. Configure it to collect and send email from your mailbox, then you will be informed about new mail pop-up window, as soon as they will come to you on the box.