There's a lot of different ways to go about it, although theres a good chance it wont taste much like the one you tasted from the meadery.. Although if you were to use honey, there's procedures i can share that could get you close
'Tart' flavors usually come from the acidity level in the wine, but using tart cherries should give you a tart-cherry profile as opposed to the sweet-cherry profile that some equate to a cough syrup flavor...
All i know is sweet cherries, as thats the primary type grown around where I'm from...
I'd shoot for at least 5lbs of cherries per gallon, although more would be better, up to probably 10-12lbs of cherries per gallon - then you're probably getting into the "pure juice" realm (which is my favorite place to be)
I wouldnt push the SG past 1.090
And if you're really going for tart, i wouldn't go above a .75 % TA
The tartness will shine, when you find the right balance between TA/pH, sweetness and alcohol levels....
Too much sugar, it'll balance out the acidity and still make a good wine but wont come across as 'tart', just 'cherry'..... Too much acidity without enough sugar, and it'll taste out of balance with a puckering sensation.. Too much alcohol, might get that cough syrup flavor and not enough, it wont keep
If you want it drier, leave the TA a little lower (.65 - .70 % TA).. Semi-sweet or sweet can handle a little more (.75 % TA)
I'm sure there will be some differing opinions..
Hope that helps, for a starting point..