I have an iPod Touch, and probably end up owning an iPhone in the future, once my cheapy flip phone's battery finally dies off. I've had my iPod Touch for a year now and I've loved it, and continue to do so.
Prior to the Touch, I was Palm fan for a few years and before that it was a Handspring. Before *that* it was my *actual* palm, writing notes on my hand or on index cards. I am and probably forever will be an incorrigible note taker, so I love electronics that help me in this regard.
But the Touch (and the Palm and the Handspring) all do so much more, from simple games to keeping me organized to capturing moments on the go, and I have become connected at the hip with these devices. (quite literally seeing as they have a permanent place on my jean pockets)
So here's a list of the apps that I find indespensible. Your mileage may vary, especially in the "to do list" category, but Apple App store has a plethora of options and different takes on this stuff.
Toodledo: This is my note taking and task list app, and I use it *constantly*. There are a lot of other to do apps out there (some appear below) but I went with Toodledo because of the price, the browser access/cloud syncing, and the flexibility. Toodledo has a lot of different options you can play with, so you can make it work just the way you like it, from super simple to full-on GTD mayhem. It also integrates with a lot of different services and operating systems, and lots of different client apps have been written to interface with it. My only quibble is that the search functionality on the iOS client only searches titles of tasks and notes, not the contents.
Honourable mentions: Paperless is a light, pretty and simple task list and note taking app; I just like the colours and fun icons. People rave about Things, and I guess it's supposedly pretty awesome, but I can't get past the price; maybe if you aren't naturally a productive person and you need a system to help you, this might be for you. Epic Win is a pretty fun idea, overlaying an RPG mechanic and fun animations over your day to day life; definitely not for power-productivity people (there's not enough options for sorting by priority, due dates, etc.), but a neat idea and potentially motivating for the right person. You might also like the very pretty My Wonderful Goals and maybe ReQall, which I haven't really figured out yet, but seems super powerful.
EDIT: A few others who might want to investigate include Orchestra (simple options on the to do front, but allows for voice-recorded to do creation and assignable to dos that come along with comment/forum style threads or chat logs, probably it's most powerful feature), Action Method (works best as project management software for co-ordinating small- and medium-sized teams, a bit overkill a personal to-do) and Wunderlist (a simple but very nice looking to do list; lacking in power-user features). All of these are free options although Orchestra and Action Method require you to create an account on their website.
Instapaper: I'm constantly coming across articles that I find really interesting but don't have the time to read at the moment. I'll bookmark them, or email them to myself, and then I still never get around to reading them. Instapaper solves this problem by allowing you to mark an article to be read later, using a little bookmarklet. The article is then downloaded by the Instapaper server, cleaned up, and then you can download it into the Instapaper client so you can read the article, uh, later, while on the go. Unfortunately I'm not commuting as much as I was before, and reading technical articles is kind of frustrating. (so many bloggers link out to libraries and demo pages, and sometimes I'd like to be at my computer so I can try out code samples) Still, it's excellent for loading up and reading while you're home for the weekend, in a line up or waiting for someone.
Byline: Some experts are saying that RSS is dead or dying, and social recommendation is the way of the future. Whatever. I still like to keep up on top of my favourite blogs using RSS feeds, and Byline is a great client. It hooks into a Google Reader account in order to handle subscriptions, and the best feature is that it can be set up to actually download and cache the whole article page, as it would appear in a browser. Very handy for RSS feeds that only spit out teasers rather than the full article. If you don't have a lot of feeds to follow (and you shouldn't), the free version should suite your needs, but it's definitely worth paying for.
Cashtrails: This is a very simple, but powerful, spending tracking app. It's not really for managing a budget, although you could use it for that. All it does is make it really easy for you to enter in when you're spending money, categorize it, and then build out some simple reports on your spending so you can see where your money is going, or where it's coming from. If the built-in reports aren't enough, you can also export the data as an Excel or CSV file so you can do your own data munging. I'm not a financial guru, and I've seen more complex budgeting apps out there that, for me, are just intimidating. Cashtrails is simple and straight forward, which means you're more likely to actually use it, which means it will ultimately be very helpful in your quest to save money.
Calvetica: The default calender in iOS sucks. SUCKS. It's so clunky to look at and slow to enter new information, it's almost unusable. Or at least you'll feel that way after trying Calvetica. This is a beautiful, super fast and easy to use calendar app. I discovered it very early on into owning my iPod Touch, and while I kind of preferred the earlier version of the app (which they released as a separate product, Tempus), the current interation is still pretty awesome and definitely a power user tool.