In NYC, you buy a metrocard, and it takes $2 off it every time you go down into the subway system. Once you're down there you connect where ever you want. The cards work pretty well and towards the end I easily shared one with my husband so we weren't trying to get two cards to their tail ends. you get bonus metrocard money the more you initially put on it.
In washington DC there's this toll system that depends on where you get on and off. you buy a card with money on it, swipe it when you go in, swipe it when you leave. the cards are cardboard and if they crease, they're done... which can be a real problem if you've gotten into the subway but now can't get out. there are some sales stations that help you, but once we were at the end of a line that was unattendend and we had no idea what our lost selves could do, so we just went out through a side gate. I learned there are huge fines for this but we hadn't figured out the system yet and didn't know what our options were, I was thinking it was like NYC where I could just swipe a card and open a gate, but if you're trying to get out someplace you can't swipe a card that doesn't know where you came in at. It got messy for us.
All in all I feel proud that I know how to get around in two subways systems now, but the fact that I've only experienced two and they're both different kinda scares me. Like, there's not one way to do a subway system, and wherever I go I'm going to have this whole new thing to learn. i'll be forever chasing different implementations. it's tough.
and yes, you can get day passes, week passes, etc. but don't forget we're trying to learn these systems as cheaply as possible. a day pass in dc was like $7... compare that to a normal subway trip which is almost always under $2, and it was sort of tough for each of us to make that up because we pretty much only went one place per day.