I'm not sure how useful this will be to your question, but what I do is not keep track of every species I've seen in every zoo I've visited, but just keep a list of all animals I've seen in total. This means I only have to worry about the common zoo species once, and because I sort this list taxonomically it means I can easily see which (sub)species of a certain genus/family/order I've seen (and thus compare with a full list to know what species I still need to see). I always include a location with every (sub)species, often of where I saw the species first, but also sometimes of where I had the most memorable experience or saw the species the best.
I've tried listing all species in a zoo by either noting it all down (but this takes a lot of time during the visit itself) or taking pictures of all signage (but this takes in a lot of storage space if you shoot in RAW and don't bother to switch over for every sign there is), so ultimately I stopped doing that and now only do so for species of which I'm not sure I've seen them (and once in Plzen for the entire zoo because I had a lot of time and had to list all passerines and primates anyways).
I'd definitely recommend making these lists in Excel as opposed to Word, though, because I feel the format lends itself to more easily sort and count things, and it's easier to spread a lot of different information like location, date, subspecies etc. over different columns.