A couple of immediate things you need to take care of soon after getting an approved Google AdSense account could be pointed out as follows -
1. Avoid indulging in too much blog commenting, Facebook share etc. Spamming may result in getting your account banned.
2. Come up with at least one post per day. I said, at least - the more, the better. Your content should always be so that it should motivate readers to visit your site again. Create useful content that proves to be valuable to your visitors.
3. It has been observed that on getting an AdSense account approved, people ask their friends and acquaintances to visit the blog and click on ads. There are termed as Fake Clicks. Never ever indulge in those acts. Google recognises fake clicks and that could be detrimental to the better prospects of your site.
4. Stick to the Google policies. Keep an eye on the updates to the policies from time to time.
5. Get a StatCounter and check the traffic your site receives from time to time. It will not only let you know the origin of your visitors, but also gives an insight into the possible fake clicks.
6. Keep your current address updated at all times. The PIN number from Google is sent via postal mail. So, having the address updated with your account is of utmost importance.
7. Don't push it to the limit. Initially, your Google AdSense revenue could be much low. It may take months to reach the minimum payment threshold of $100. Having patience and keeping up the site well updated should be your mantra.
8. Always interlink your content. It will help gain more visits - especially to your old content.
Live....and Let Live!