1. I want to use my existing Microsoft Outlook to connect to the business email account
Setting up Outlook
Install Google Apps Sync for Microsoft Outlook® from here
Additional resources are available here
I want to send email using an alias on my account
See the answer at Send from a custom From address for information.
2. I want to access my business email from anywhere
You can access the business email from any computer using the browser (Chrome, Firefox, Internet Explorer, Safari). Browse to www.gmail.com and log in with your email address and password.
3. I want to access my business email from my mobile device (iPhone, Android Phone, etc.)
Instructions for Andriod phone
Instructions for iPhone
4. I want to set up Outlook to connect to a second Gmail account
You can connect to multiple Gmail accounts. However, the main account that Google Apps Sync for Microsoft Outlook® connects to will be the only account to sync your calendar.
Step-by-step guide to link to another Gmail account at this site.