Welcome aboard, adnium - glad to have you along for the ride!
OmniFocus integrates with a free piece of software called
Growl to present pop-up alerts. A task becoming available is one of the events it can notify you about.
Install that, and you'll have the pop-up notifications. Once the start date for a task moves into the past, the Growl notification appears. (Open the Growl Preference pane, switch to the Applications tab, and double-click OmniFocus in the list that appears if you want more control over the alerts.)
Re: a today list - I helped a previous switcher use OmniFocus' Perspective feature to set up a list similar their Today list. Here's
a link to that thread in case it helps you, as well.