Define and give examples for "Bitcoin client" please. You mean like Coinbase? What about
https://blockchain.info/? If so what is considered the "safest" one these days?
I already have a BTC-e account and BTC deposit address so I'm good there
I don't know if they're better, but I prefer using desktop Bitcoin clients.
Take a look at Bitcoin-qt or Electrum.
Electrum might be a lot better for you because you won't have to download the entire blockchain
https://electrum.org/