I'm new to this too... but because I just have an itouch, not the phone, and haven't managed to work out why my router won't let me use it to connect to the internet (

) I'm using non-web based apps.
So far my absolute favourite is 'Things' because I use it as a 'to-do' list every day. It's really helped me organise myself.
I also love the 'notes' which came with it and I use it a lot. Oh, and I also downloaded an excellent 'contacts' app (address book)
For distraction I downloaded the book 'Benjamin Button' because I've been travelling a lot recently and I have really enjoyed it.
For pure escapism I also downloaded 'Flash Gordon' comic
I have never played computer games in my life but I downloaded this little game with furry balls that you ping about (can't remember the name) which is quite challenging for my brain!!! I like the fact it's a puzzle more than a mindless game. I also tried 'Donkey' but once you've done it once it's pointless so I've deleted it.
I'm looking for more suggestions too...