The first time I synced my iPhone with iTunes, all my email account settings were synchronized, and I had no problem receiving messages from, for example, my IMAP, MobileMe, and Gmail accounts.
For accounts that provide "push" services (notably MobileMe and Exchange accounts with ActiveSync), your messages can be delivered immediately or, at your option, fetched at regular intervals or manually (turning off push improves battery life), which is the only option for accounts that don't support push. With or without push, almost everything worked as I expected it to, including filing read mail in IMAP mailboxes.
But one UI choice in the iPhone's Mail application bugs me greatly. In one of my IMAP accounts, I have a lot of nested mailboxesfor example, a Take Control mailbox with sub-mailboxes for each of the books I've written in that series. In OS X's Mail, Entourage, or any other IMAP client I've used, I simply keep those higher-level mailboxes closed until I need them; then I click a disclosure triangle or + sign next to the topmost mailbox to expand it and show the mailboxes inside.
Alas, the iPhone's Mail app has no way to "collapse" mailbox lists. Sub-mailboxes are shown indented, but there's no way to show just the top level of the outline. That means if I want to move a message into a mailbox later in the list, I have to scroll past many mailboxes that I'd normally never see.
This wouldn't be a problem if the iPhone's Mail app let me selectively subscribe to IMAP mailboxes, as Entourage and most other modern IMAP clients can do. Apple Mail on OS X also lacks this capability, and I've always found it to be an incomprehensible omission. Similarly, neither app lets me reorder the mailbox list arbitrarily.
In order to work around this problem, I may have to give some thought to reorganizing my IMAP mailboxes so that the ones I access most frequently are earlier in alphabetical order (and thus appear closer to the top of the mailbox lists). But it's silly that I should have to do thatApple should be more clueful in designing their email apps.
Finally, the iPhone has no spam filtering. Although some of my accounts have excellent server-side spam filtering, others don't, and for those I rely on a smart client-side tool like SpamSieve.
But in the absence of such a program on the iPhone, a lot of those messages that conveniently appear on your screen (using up battery life and bandwidtha concern for those with caps on data transfer, like Orange's 500 MB monthly limit here in France) will be spam, and will require extra work to delete manually.
Web and Network Access
The included Safari Web browser renders pages well, and both zooming and rotation work as expected. I was surprised, though, to see that a number of sites loaded significantly more slowly than on a Mac, even when using the same Wi-Fi connection.
I would have liked to see support for form-filling and password saving comparable to what's in the OS X and Windows versions of Safari, or better yet, an iPhone version of 1Password .(1Password can, after a fashion, sync passwords to your iPhone using a special Safari bookmark, but the forthcoming my1Password service is what I'm really looking forward to.)
I set up the built-in VPN client to connect to my WiTopia account using PPTP, and it appeared to work without any issues. However, activating VPN requires a few taps, and it doesn't necessarily reconnect automatically when you switch networks (say, from Wi-Fi to 3G).
So if VPN access is a necessity for you, you have to get in the habit of looking for the little VPN icon at the as you move from place to place to make sure the secure connection is still active.
Battery Life: The Hard Warmer Option
As has been widely reported, the battery drains pretty quickly. I haven't yet spent enough time with the iPhone to see how long it takes to get all the way to zero in normal usage, but my casual observation is that the combination of GPS, 3G, Wi-Fi, push email, and music playing tends to make the battery level fall at an unpleasant rate.
Apple recommends turning off power-intensive features (the ones mentioned, plus Bluetooth, the equalizer on the iPod, and a few others), but of course that means you're also turning off the device's most interesting and useful features. (Heck, I can make the battery last for days by turning the iPhone off and using it as a paperweight, but that's kind of missing the point.)
I have noticed that when the phone is busy doing processor-intensive tasks like playing videos, the back gets quite warm. Not lap-searingly hot like my MacBook Pro, but certainly keep-your-hands-toasty-in-winter warm. I think Apple should really advertise that hand warmers are among the many pocket gadgets the iPhone can replace.