Tin will darken with age and use. That is normal; eventually it becomes a medium dull gray.
Don't scour your pots, the tin will scratch and come off and will need re tinning before its time. Use wooden and nylon utensils.
Wash pots with dish soap and a sponge or soft rag. Let them soak if necessary to loosen stuck on food. Minimize this need by emptying pot into a serving dish and letting it soak while you eat. Or put pans to soak while having coffee and desert.
The copper outside can be cleaned with any copper polish or Bar Keepers Friend. Regular polishing keeps it shiny easily. Many folks like the "used look" of a dark penny color. The pot will still cook fine whichever you choose.
Your pot should not be heated without food in it. Tin melts at 460* Get to know your pan's characteristics. Boil an egg, make mashed potatoes, cook oatmeal or rice, etc.
All that said, restaurants that use copper pots (and many do) put their pots n pans through a lot of "abuse" from the home cook's view point. But they also send them out for re tinning every year or so. (the equivalent cost of replacing your hotel pans and pro aluminum sautees every couple years after they get beat up)
I've been cooking in tinned copper for about 10 years at home and have yet to send anything out for re tinning. I have 3 great contacts for such work when the time comes (Denver, Newark, Brooklyn, but all I've done with them so far is buy some hand hammered pots! (just what I needed)
check out Fante's Kitchen Wares Shop - fantes.com
and read up on their cookware section about various metals and specifically copper and it's care.