There are several ways to find out about new cases that need your attention.
FogBugz sends you email to tell you when a case needs your attention:
FogBugz does not send email notification to tell you about things you did yourself. If you assign a case to yourself, you will not receive an email. If you send in an email to a FogBugz project for which you are the primary contact, you will not receive a notification for the case created.
At the bottom of every case page is a link that lets you subscribe to a case you're interested in. FogBugz will send you an email notification whenever anything changes about that case.
In the interest of reducing the amount of unwanted email, there is no way to force someone else to subscribe to a case. If you want them to know about it, you should temporarily assign the case to them with a note like "FYI" (which sends them a notification email) and then immediately assign it back to whomever it really belongs.
In the main list, FogBugz shows cases that you've already seen in purple. Cases you've never seen, and cases which have changed since you last saw them, are shown in blue. This is tracked by FogBugz, not your web browser.
You can use the special search term show:unread to search for cases that have changed since you last saw them. Either:
FogBugz publishes RSS feeds, allowing you to use an RSS aggregator to receive notifications, so you can keep up to date on changes to your filter without opening your web browser.