Wash the tank, definitely. It's good to wash them on a regular basis anyways. Just use warm water and dry it thoroughly (q-tips and paper towels are your best friends)
As for the coil, that depends on how many flavors you use and the types of flavors. If they are compatible, then I personally just use the same coil as is. When I use strong flavored juices, such as menthol, and I know I'll be using the same flavor again, I put the coil aside for re-use with that same flavor (you can use a small ziploc bag and write the name of the flavor on it.)
Washing out strong flavors can get a tad complicated.... they often can stick for a while even after dipping in alcohol for a day or two. My personal coil washing technique is:
-Put it in a small glass and wish it around in warm water. Empty, rinse and repeat at least 3 times.
-Let it sit in warm water for a few hours
-Put alcohol in the glass (cheap vodka is strongly recommended for this), swish it around and let sit 24-48 hours. Swish it around every now and then
-Rinse the glass and repeat step one a few times
-Let it dry on the side for about 24 hours or more.
By experience, menthol, mint, cinnamon, chestnuts and similarly strong flavors will still take a tank or two to disappear. For that reason, I use those coils with other strong flavors.
If you have the BVC coils in your Nautilus mini (extremely likely), you can also "dry burn" your coil. This is achieved by mounting only the base of the tank with the coil to your vaporizer and holding the fire button for 1-2 seconds at a time for about 5-6 shots. I suggest doing that before cleaning the coil and doing it again, but conservatively, after the cleaning process. No guarantees that will eliminate strong flavors 100%, but it will make a difference.
Cleaning and dry burning can also extend the life of a coil as they will gunk up after a while or when the taste is a little off.
Alternately, I just found a "buffer" juice that I like. It has a very neutral taste that mixes well with other tastes, so I use one tank or two of that juice until the original taste is gone. It is better to get a second tank or more to use that technique, though. I decided to forgo that whole process and I'm up to 9 tanks now as I discover too many juices in my travels to be able to spend all that much time rinsing and cleaning coils.
And welcome to the forum!
Hope this helps,
The Torch