Well, there are different trains of tought on the best option... Mixing is defenately the easyest option of the 3... But you'll need to put your trust in the mixing service... If i'm not mistaking Theymos (The admin) isn't a big fan of mixing services for this very reason... What if the mixer is a scam or a honeypot? What if they made mistakes in their security protocol, what if their servers get seized?
But, if you trust them, a mixer is a cheap and fast method

The 3th option isn't that hard either:

You controll unspent outputs funding your address, let's call it 1BtcAddressA. At this point, you go to an (on-the-fly) exchange and exchange your BTC to (for example) XMR (or ZEC or Verge or...). You create an XMR  wallet on your hardware wallet (or desktop wallet) and withdraw the XMR to your local wallet (you now have an unspent output on the XMR network funding address 4XmrAddressB). Since this is a high anonimity coin, only the exchange knows they funded 4XmrAddressB. Now, you either create a second new XMR wallet (or a new address within the same wallet if you trust yourself not to start mixing unspent outputs) and you use the unspent output funding 4XmrAddressB to fund 4XmrAddressC. At this point, the link is completely broken.
Now, you use a different exchange to exchange the unspent outputs funding 4XmrAddressC on the XMR network for an unspent output on the bitcoin network funding a brand new address on a new wallet (1BtcAddressD).

At this point the link is completely broken. If you're paranoid, you can perform these steps over Tor and VPN, or you can mix several methods just to be sure...

Edit: it's friday, i'm bored, i added a visual aid... Copyright: Me