Question : Surf internet with GSM 3G/HSDPA unlocked quadband phone

If I had  a phone with the above-named features and a slide-out keyboard, what else do I need  in order to surf the internet

1. in a Wi Fi hotspot
2. elsewhere

The phone has Windows Mobile 6.1 operating system.
I have not yet purchased the phone. Would I be better off with a different operating system?

Answer : Surf internet with GSM 3G/HSDPA unlocked quadband phone

1. WIFI: I could use this phone to go to my favorite websites while sitting at Starbucks.  I would not need to download anything before my first surf because the phone's operating system has a browser built in. I COULD SURF EVEN IF I DID NOT PUT MY T-MOBILE PRE-PAID SIM CARD IN THE PHONE. If I did put the sim card in the phone, I could make voice calls.  

You are correct. It depends on the phone whether or not it can be used without a sim card. Even if the sim isn't activated, some phones (HTC 8125 for example) only operate in emergency mode without a sim.
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2. NON-WIFI:  My question did not concern using the phone as a modem connected to a laptop, but rather browsing using the screen and keyboard of the phone itself. Can this be accomplished without purchasing a long-term contract from a cellular service like T-mobile?

As far as I am aware, the sidekick data plan is the only available data plan with T-Mobile prepaid service. I did see, however, a forum post on T-Mobile's website which says with a little tweaking, you can get T-Mobile SideKick data to work on any phone:

http://forums.t-mobile.com/tmbl/board/message?board.id=Prepaid&thread.id=322

With AT&T you can add a prepaid unlimited data for about $20/month extra,  It might do you well to consider switching. I am sure if you talked to AT&T they would give you input on how to keep your existing phone number(s).