Using Filters in Gmail

To move certail mails directly to specific folder bypassing your Inbox, Gmail filters can be used:

• Click Create a filter.
• Enter the desired criteria.
• To filter all mail from specific email id, type their email address in the From: field OR to filter all mails with a specific subject Line type in the words in the "Subject" field (This option is very handy if you want to filter mails received from groups you have subscribed as mostly all such mails have the Group name appended to the Subject)
• Click Next Step ».
• Check "Skip the Inbox"
  
• Also check Apply the label:

*Select an existing label (folder) from the Choose label... menu or:
*Select New label...
*Type the desired name for the label (folder).
*Click OK.
*Optionally, check Also apply filter to conversations below to move existing messages matching your criteria to the folder.

• Click Create Filter.

Transfer emails between two Gmail accounts

Here's how you can transfer all your emails from one Gmail account to another:

Step 1: First, log into the old email account (account you want to import the emails from) and click on Settings, then Forwarding and POP and choose Enable POP for all mail. Click Save Changes at the bottom.

Step 2: Now log into the new email account (account you want to import the emails to) and click on Settings. Then click on Accounts and click on Import mail and contacts button.

Step 3: Type in the old Gmail address in the text box and click Continue.

Step 4: Now type in the password and click Continue.

Step 5: Type in (if its a text box) or select pop.gmail.com from POP server dropdown list. You can also select Port 995, click on "Use SSL" and click "Continue".

Step 6: Select the options you need and click on "Continue"

Step 7: Now wait and watch all your emails getting transferred to the new Gmail account. If you have lot of emails it may take several hours for the emails to get transferred/imported.

Once the import has started you chan check the Import status/history in "Settings >> Accounts >> Get mail from other accounts >> View History".
Also there is no need for you to stay logged in once the mail transfer/import has started.