First off, you NEED a Google email account to use the Market and download Apps. So you're going to have to set one up.
Now as far as email goes, Android phones have an app installed specifically for Googlemail and if you are used to Googlemail on the web then that's great - stars, labels, etc. Me? Never used Googlemail on the web, so I installed K-9 Mail (free on the Market) and set up my Gmail account as an IMAP.
Google's IMAP servers support IMAP PUSH (IDLE) so I am getting instant email with my Gmail account using K-9 Mail. Sent items are sync'd and as I also use IMAP to access me Gmail on the desktop, it all stays in sync.
I also have a Hotmail account and I have set up K-9 Mail to access this via POP3 and check for email every 15minutes. My Hotmail account is my secondary account so four times per hour is fine to check. K-9 leaves the mail on the Hotmail servers so they are there when I check later on on the desktop. Only problem is that sent items aren't sync'd when sent from the phone.
Lastly I have a number of my own domains and these are hosted at 1&1 and I have set these up using IMAP as well. 1&1 IMAP servers also support PUSH (IDLE) so I get instant email on those too.
So that's all my email, mostly instant, without having to use Google's mail App.
As far as contact and calendar goes - I tried letting the phone sync with Google's online contacts and calendar but hated it. Can't get used to using the online contacts and calendar and their crippling lack of functionality so ended up installing HTC Sync and now I sync my handset to my desktop Outlook once per day.