I am new and I did not get bitcoins I am new to Bitcoin. So, I downloaded the Windows Bitcoins software from the official website and used the "request bitcoins" option to request bitcoins from my exchanger.

My exchanger charged me $20 and told me he will send me bitcoins corresponding to $20. But even after 18-20 hours I didn't receive any bitcoins.

Additionally, the Bitcoin software is showing synchronizing with network : x years xx weeks remaining. 

What should I do? Should I wait for the synchronizing to complete or file a complaint against exchanger?
Yes, wait for it to finish synchronizing. You will not see your coins until it finishes.

(Note that the time shown is not the estimated time until completion; it represents the age of the old transactions currently being downloaded. Depending on the speed of your computer and internet connection, you can expect it to finish in a few hours to a few days.)

In the meantime, you can enter your address in a "block explorer" site like http://blockchain.info to check whether the seller made the transaction as agreed. (This is safe; your address alone cannot be used to steal your coins.)

If you don't want to download the entire block chain, you can use a different client such as Multibit, but you will have to import your wallet.dat from Bitcoin Core into your new client software. Its documentation should explain how to do that.