Anyway, I would usually do my villager requests as soon as they ask because I would forget them later on after I'm done playing. I do them later sometimes if I want to play again.
I say yes to everything even if I know I'm not going to be able to catch a specific bug or fish (not sure whether not doing a request or denying a request lowers friendship more??) but I try and do everything else right away or else I'm certain I would forget. Sometimes I jot down what everyone is looking for if I get several in a row and I'm in the middle of doing something else. If I agree to meet someone at their house or my house at a later time, I usually set a reminder on my phone, because those I definitely forget.
I usually do things right away, since that's easier. The exception would be if there's a furniture request, because I might have to order something to suit their house. I'd probably forget otherwise.
If they ask me for fruit (if it's native or already in my town) or to deliver something to someone, I'll do it immediately. But if it's a fish task, I do it later (most of the time never) and if it's a bug task, I'll say yes but not do it at all unless I come across it.
If I'm able to I'll do it straight away but as soon as it's a gift to another villager, the one who's getting the gift usually seems to have mysteriously disappeared.
If it's a delivery I keep saving and quitting until they go in their house. If it's something ugly then I'll keep resetting until they give it to me lol. I'm pretty OCD at keeping my villagers houses looking decent.
I try to do it right away so I don't forget! If it involves catching a bug or fish that's only available at a certain time of day, I'll time travel so I can get it done as soon as possible
if it's a delivery or a fruit run, i'll do it right away before i forget or accidentally open the present, but if it's for furniture or a bug / fish, i usually save it for later. i'm awful at fulfilling furniture/bug/fish requests though haha, partly because i don't want them to turn their houses into museum exhibits
I usually do them asap, unless I have to deliver something to another villager. Usually the villager I have to deliver something to ends up being in Main Street, and because I'm too lazy to go look for them, I give up and forget I had to deliver something. Oops lol
This usually happens: if it's a request where I have to give a gift to a villager, I'll scope out the villager to give it to them asap. If I can't find them immediately, I will most likely forget because I don't like hanging around to find them (now a days I don't play very long) If it's fruit I'll get it for them straight away because I usually have fruit in my pockets.
I used to break my neck trying to fulfill villagers requests. Now they're started asking coelacanths and stuff and I'm way too tired for all of that LOL. I hate when they ask for random rare stuff.