Trading

see wallets under Tools

buying and selling

LocalBitcoins - bitcoin escrow service and face-to-face meeting platform.
Bitcoin-Treff.de - buy bitcoins face to face without fees or id checks in germany.
Mycelium - mobile bitcoin wallet with 'marketplace' function allowing face-to-face meetings (with fees)
Bisq - The P2P exchange network a peer-to-peer exchange with no centralised server and no fees!
Crypto-to-crypto: bittrex, shapeshift.io, binance, flyp.me

Commercial exchanges

Note: These are links to some known exchanges. Being listed here does not imply they are recommended.

kraken well established european exchange.
bitstamp Luxembourg based exchange.
coinbase Very popular San Fransisco based exchange.
coinjar Australian based exchange