Do I need both bitcoin client and guiminer to do mining? I am using bitcoin for the first time. Do I need to run both bitcoin-qt and guiminer to start mining? I have installed both programs and currently bitcoin-qt is running and synchronizing with network and there are around 60 more weeks to go.  

So I guess once synchronizing is done I will get a wallet? And then I can run guiminer so I can start mining in "Slush" mining pool? (I have already registered for an account and worker)  

And once synchronizing is done I can close bitcoin-qt and then just use guiminer?
If you are mining at a pool, you don't need a client to be running - the pool will provide you with all necessary information.

If you are mining by yourself, you need to run the client to know what you need to put in a new block.