Ok, you've backed everything up from your old host, set up an initial site on your new host, and are ready to do the transfer of the first domain.

1. Note Old IP Address: I ping both the base URL (ie: CoolTechU.com) and the mail server URL (ie: mail.CoolTechU.com) and note the current IP address each ping resolves to.  You'll monitor your change based on these addresses in step 3, below.
2. Transfer the DNS: This is where I hold my breath.  This is the point where, if things don't work out quite right, e-mails can drop out of site, first-time visitors to your site could end up believing your site is gone forever and never return, and you can regret your decision to switch in the first place.  But usually it isn't that bad.  From your new host, note the DNS (domain name server) URLs and IP addresses.  Normally the host will provide you with two -- a primary DNS server and a backup, secondary server.  Usually, the URL (not the IP address) is all you need.  Go to the manager of your domain (mine is GoDaddy.com) and change the DNSs currently assigned.  Once you save your changes, the process is underway.  In effect, the change propagation begins immediately, making its way throughout the entire Internet.  You are normally warned that this could take from 24 to 48 hours.  Although this was true a few years ago, it seems to be a leftover CYA statement.  I've seen the change happen within 10 minutes at times (I check from a few different DNS servers to confirm), and within a few hours at most.  It's a lot faster than it used to be.
3. Monitor the Change: I normally do this in a few steps.  First, I ping both the base URL and the mail server URL, comparing the IP address to what I noted in step 1, above.  I try to do this from machines from at least three different networks.  Not everyone will have this luxury.  I have both DSL and Cable services at home, so I tested on machines hooked up to each of those networks.  Then I VPN into work, and do the same ping test from my machine there.  Once I see the base URL resolving to the new IP address, I post a "Temporarily Down" default page to the website of the old host, if I hadn't yet done that.  At this point, I need to FTP into the old host using the old IP address, since the site now effectively points to the new host.
4. Test the Website:  At this point, I'm ready to test the site from the new host, but I may still need to change back any temporary website configuration settings I made when testing the site after uploading using the temporary URL. When those changes are done, I then browse to the site using the normal URL.  The "Temporarily Down" page should not show up -- the page should open correctly.  Then I know for sure that the base URL change worked fine from that network.  Then I try it from the other two machines.  Your mileage may vary -- if you aren't able to test this from multiple networks, play it safe, and wait a few hours before doing the e-mail transfer test.
5. Test the Mail Server: If the ping tests in step 3, above, did not yet show the mail server change, I re-test that.  Once the change shows up on all three networks (again, if you only have access to one network, wait a few hours before doing this next step), I send myself a test e-mail from my old host's webmail application.  If I receive the e-mail on my new host, it looks like the mail server transfer is complete.  Then I change my mail client's settings, if necessary.  You may not need to change anything here, if the pop and smtp accounts are the same (they are usually mail.MyDomain.com for both).  Often, I'll use a different password for my e-mail account at the new host to help augment this next test -- this is because when I do the next tests, if I get an authentication error, I know right away that I'm still hitting the old server.  Then, I CLOSE and re-open my e-mail client.  I use Outlook, and it appears that Outlook cache's the old IP address that the URL resolves to, so it may still attempt to hit the old host's server. Other clients may do the same thing, but I'm not sure which.  Neglecting to give this instruction to the users of my servers is the cause of more questions and problems than any single other thing after a transfer.  Then I send myself a test e-mail from two different accounts.  If I receive those fine, I reply to each of these messages.  If that works fine, then I know this test is complete.
6. Setup Other Services: Now that the two main pieces of the domain are working, if you have any other services that you need to activate (such as SharePoint), then do that.  My sites are pretty basic right now, so I'm normally done with this initial domain at this point.
7. Setup Aliases or Domain Pointers: If you have multiple URLs pointing to the same site (ie: I have MarkFreedman.net and MarkFreedman.org pointing to MarkFreedman.com), set this up next.  Each host has a different way of doing this.  Some simply allow you to create an "alias" to an URL, where you would simply enter the URL to point to it.  Other hosts will require you to add a second domain with this URL, then on the control panel section for maintaining a website, allow you to "point" that second domain to the website setup under the first domain.
8. Test Alias Access: Now that the aliases are setup, I test the site and e-mail accounts, making sure all results are equivalent to the main domain.
9. Setup More E-Mail Forwards: I got burned with one of the hosts because of the following: Normally, when you have multiple alias domains pointing to your main domain, e-mails will also get directed to the main domain.  In other words, for e-mails going to my MarkFreedman.com account, if anyone sent anything to MarkFreedman.net, it would still get to the .com mailbox.  But this did not work for my current host.  I had to add e-mail forward accounts at the .net domain to point to my .com addresses.  This is a very annoying pain, because for one of my domains, I had five times as many e-mail addresses to create.  At least forwards are simpler.  They don't need passwords, etc.
10. Notify Users of the Domain: At this point I'll put together individual peronalized e-mails to the alternative address of each person who has an account at this domain, telling them about the switch, explaining the reasons for the switch, giving them their temporary e-mail passwords (and FTP account info, if necessary), remind them to change their password through the new webmail system, point them to instructions on how to use the new webmail client, and give them instructions on how to access their old webmail account in order to clean up any messages still sitting there.  I usually give them a deadline, because I need to cancel the old account as soon as possible.  If they don't meet that deadline, that's when I go into their old accounts myself, and forward the messages to their new accounts.  I warn them that I need to do this, before I actually do.
11. Delete Content from Old Host: Even though I'm closing the old account, and deleting the old websites / domains through the control panel, I first take the extra, precautionary step of deleting all the files via FTP first.  That gives me a sense of control, even though they can easily have a backup already.  Of course, if you have any real sensitive data, it should always be encrypted on the site or the FTP to begin with.
12. Delete E-Mail Accounts from Old Host: Similarly, I manually delete all the e-mail accounts from the old host before deleting the domain.  Do whatever you feel comfortable with.  I'm always extra cautious, and manually delete the accounts (after everyone had a chance to check and forward any remaining e-mail, of course).
13. Delete Old Website and/or Domain: The control panel should have an option for deleting a domain.  If it also has a separate website option, then delete the website first, then delete the main domain, and each related "alias" domain.
14. Transfer Remaining Domains: Well, one domain down, six (for me) to go.  When you feel comfortable that the first transfer is working fine, lather, rinse, repeat.  Isn't this fun???

In my next post, I'll continue my hosting nightmare story.  I was with six separate hosting companies in two weeks.  Yep.  Fun.

My Web Hosting Nightmares, Part 1 - Background
My Web Hosting Nightmares, Part 2 - Preparing for the Transfer
My Web Hosting Nightmares, Part 3 - Starting With a New Host 
My Web Hosting Nightmares, Part 5 - My Continuing Story