Yes the iPad screen gets quite grubby quite quickly. Best not to get too hung up about it - it's gonna happen however much you use the device. I give my screen a polish every few days with the following products:
iKlear Apple Polish - This is pricey but worth it - I've just bought a new kit after I ran out of fluid on the last one and that lasted 6 years! I use it on everything - iPods, our Macbook and iMac, the TV, the Nintendo DS. Leaves everything glassy smooth and smear free. The microfibre cloth that comes with the kit is of very high quality and helps get your kit back to pristine.
Not all microfibre cloths are the same - the closest I've found to the quality of the iKlear cloth on the cheap is
this (yellow duster) from Lakeland.
Also from Lakeland is the
E-cloth which works with a little spritz of water on the equipment - I'm talking a fine mist, achievable with a
Muji pump spray. This leaves the screen smear free also, at a fraction of the cost (well 1/4... which is a fraction

) of the iKlear kit.
Some of the case prices can be pretty astounding, so I opted for
this from Amazon. Cheap but looks the business. Although the case has a built in stand making it ideal for movies, even internet, it is too upright for typing.
As for Apps - for gaming try AirAttack HD - on the app store for 59p and a fairly decent old skool shoot 'em up. Osmos HD is a slow but beautiful strategy game. Seek out Bebot - basically a sound generator but multitouch and (relatively) beefy pad speaker make this muchos fun.
Anyway, enjoy your iPad