Any web site should be tested for cross-browser and cross-platform compatibility -- this is a primary rule of web site quality assurance. However, wait on the compatibility testing until after the site can be verified to just plain work. Test the site's functionality using a browser/OS/platform that is expected to work correctly -- use what the designers and coders use to review their work.
By running through the site or application first with known-good client configurations allows testers to focus on the way the site functions, and allows testers to focus on the more important class of functional defects and problems early in the test project. Spend time up front identifying and reporting those functional-level defects and the developers will have more time to effectively fix and iteratively deliver new code levels to QA.
If your test team will not be able to exhaustively test a site or application -- and the premise of this essay is that your time is extremely short and you are testing without a formal plan -- you must first identify whether the damned thing can work, and then move on from there.
Concentrate on Ideal Path Actions First
Ideal paths are those actions and steps most likely to be performed by users. For example, on a typical commerce site, a user is likely to:
identify an item of interest
add that item to the shopping cart
buy it online with a credit card
ship it to himself/herself
Now, this describes what the user would want to do, but many sites require a few more functions, so the user must go through some more steps, for example:
login to an existing registration account (if one exists)
register as a user if no account exists
provide billing & bill-to address information
provide ship-to address information
provide shipping & shipping method information
provide payment information
agree or disagree to receiving site emails and newsletters
Most sites offer (or force) an even wider range of actions on the user:
change product quantity in the shopping cart
remove product from shopping cart
edit user information (or ship-to information or bill-to information)
save default information (like default shipping preferences or credit card information)
All of these actions and steps may be important to some users some of the time (and some developers and marketers all of the time), but the majority of users will not use every function every time. Focus on the ideal path and identify those factors most likely to be used in a majority of user interactions.
Assume a user who knows what s/he wants to do, and so is not going to choose the wrong action for the task they want to complete. Assume the user won't make common data entry and interface control errors. Assume the user will accept any default form selections -- this means that if a checkbox is checked, the user will leave it checked; if a radio button is selected to a meaningful selection, the user will let that ride. This doesn't mean that non-values that are defaulted -- such as the drop-down menu that shows a "select one" value -- will left as-is to force errors. The point here is to keep it simple and lowest-common denominator and not force errors. Test as though everything is right in the world, life is beautiful, and your project manager is Candide.
Once the ideal paths have been tested, focus on secondary paths involving the lower-level functions or actions and steps that are less frequent but still reasonable variations.