Being an avid user of Gmail and Google Apps myself, I know that it's not as simple as it seems, there's a few steps involved.
You have to redirect all of your work emails to your personal one, verify that you own your personal email, verify (again) that you own your work email so you can send mail through that address and setup a filter so that things don't go into super confusion mode. Long!
Thankfully for you, I'll tell you exactly how you can acomplish this! I'm so nice.
Follow these steps:
Login to your work email account
Go to Settings on the top-right corner of your screen and select Forwarding and POP/IMAP -> Add a forwarding address
Enter your personal email, along with a tag, such as: username+business@gmail.com where business would be the tag that you associate your work emails to (although this can be anything, make it easy to tell what it is)
Login to your personal Gmail account
Go to Settings -> Filters -> Create a new filter
In the To: field, put the address you chose for forwarding your business mail to (in my example, username+business@gmail.com and, on the next step, choose to apply a new label (called Business for example)
Still in Settings, go to the Accounts tab and Add another email address you own
Put in your business email (lets say username@company.com and also specify the same address as the reply-to address and you can choose to use either SMTP server (I usually don't bother)
Accept both authorizations sent to either emails and you're done!
When mail arrives at your work box, it will be redirected to your personal box but with whatever label you chose (in the example, Business). Makes it much simpler to organize your mail in a way that won't consume your time searching for it.
Hope this helps!
First problem
Google treats the retrieving of external POP3 accounts using their own algorithm and that says in sum that, if you get many email, it will fetch with small gap times, if you have emails once upon a time it will expand the recurring time as there is no need.
You should avoid this completely if you need that POP3 emails quick, for that, you need to create a simple FORWARD email that what it does is
every time I receive an email, I will forward to the address xpto
from your Control Panel, just do this:
This will make sure that you do receive as fast as you can (as it's at the same time it reaches the POP3 server) and if you configure your GMAIL to filter all emails that were sent to your whateverdomain.com you can easily flag them and even forward again, but this time using GMAIL.
Second problem
You can have multiple accounts on your iPhone, I have several as the image below shows:
Just add your Gmail and your POP3 Account.
Third problem
If you use a Forward account, you cant set up a new POP3 account, as it's not a POP3 account anymore and you are not keeping anything on whateverdomain.com just in your forward email.
If you need to keep things in the POP3 Account as well, I would suggest that you create a new POP3 account and on GMAIL (that receives the forwarded Email), create a filter to Forward it again to the new POP3 account (or a new GMAIL account just for this, for example as the basic storage size is always a little bigger than regular POP3 account sizes).
Something like this:
Then in your Mobile phone you can use only GMAIL (as the messages are being dropped into your personal account) or set up the POP3 account and you have one account to send and receive emails just for the whateverdomain.com
Best Answer
For individual messages, you will need an IMAP capable email client and connect it to both (or more) accounts. I can only speak about Thunderbird, but the same principles apply to other tools as well.
Basically, either copy or move the message from one account to another.
Beware, however, of folders! GMail has no such thing as folders, but only tags. Thunderbird translates this into folders. One GMail message with two tags corresponds to two (identical) messages in Thunderbird, in different folders.
If you move a message out of a folder, Thunderbird essentially tells GMail to remove the label from the message, which is NOT the same a removing it entirely. You may not care about residue in your 'All Mail' archive, but if you do, I found only one way to get it done:
NB: I never tested what happens if TB implements a move (as it does) as a subsequent 'copy' and 'move to Trash'. In GMail, messages live in the Trash for 30 days. What happens 30 days after a simple move from a folder in one account to another account, I don't know.
Another option is to 'bounce' messages. This fits the use case where you receive e-mail in one account, but should really have received it elsewhere and plain forwarding undesirably changes the message headers (sender, receiver, reply-to, dates, reply-to, etc). See this question for more details.