Where does the money go? Is it possible for any person to setup his wallet application to process transactions and make some more BTC?
If you run an Escrow service, yes, you could get paid to process transactions. But that's probably not what you're asking about.
In normal cases, transaction fees go into the reward for a new block being mined and are earned by the miners that create the new block. This takes specialised hardware and software. If you want to get your feet wet in this area, you ought to look into mining at a pool.
When you are "mining" BitCoin, you are processing (providing 3rd party confirmation of) transactions.
Each Bitcoin block is a list of recent transactions received over the P2P network, from bitcoin clients. Consider it like a double-entry accounting ledger of some transactions, with particular focus on their exact order, based on timestamp. A miner adds a special generate transaction to the end of the block and then uses a Cryptographic hash function to calculate a candidate hash value for the block. If the numeric hash value fits under the "limbo stick" of current Bitcoin difficulty, the network considers the block valid, awards the miner the 50BTC reward and any transaction fees, and the transactions within the block receive a confirmation.