Finally synchronized the blockchain, yet not sure if I'm mining for coins. Where do I find out if I am or do I need something else?
The bitcoin wallet is for receiving and sending bitcoins. When it is synced you are all set. You don't need anything else to receive and send coins.
Note that you can receive coins even if the wallet is not synced, you just won't see them before you are synced.
I assume that you have downloaded Bitcoin-QT, the standard bitcoin client. This enables you to manage your addresses, to send and to receive Bitcoin.
If you want to mine bitcoin you will need special software, and if you want to mine bitcoin profitably you will also need special hardware. There is numerous questions about "how to mine with X" on this site, if you want to get started with that. See for example: How can I start mining Bitcoin?