I suggest you get a deterministic wallet for many reasons:  your privacy, ease of backup and long term fungibility of the Bitcoin system.  I use the Trezor wallet but with a little research you can find other deterministic wallets.

Deterministic wallets never reuse the same Bitcoin address.

Now about your other coins.  Just send them to a trading account somewhere (Bitfinex, localbitcoins, any account will do) then wait a while, say a few days, then transfer your balance back to your brand new Trezor or other deterministic wallet.  You will not get your same coins back from the exchange.  When you deposit them they go into their big accounts for trading, when you transfer your balance out you will get different coins.

Or to more instantly exchange your current coins for new ones use bitmixer.io.

Get your deterministic wallet all set up and then send your BTC to bitmixer.io and for a fee they send you back new coins to your new wallet.

From then on always make deposits to your wallet onto the next address given to you by the wallet (the next one in the sequence).  Do this every time.

That is a good start.