You have access to the save screen at the end of each day.
As for the game running slow, that's mostly just the game engine itself, you'd need a US DOD-level supercomputer to force it to really run smooth. Load times can be dramatically shortened by using a solid-state drive for the game (and by dramatically I mean the opening scene where you introduce yourself to the class goes from a 2+ minute wait to a 10-ish second wait) but the game has to stop and load a student every time someone walks into the worldspace ("room") you're in, and when you load into each scene it's technically not done loading until the audio happens, because the game engine was apparently entirely coded one-handed. (You'll notice there's a period of time where everything's silent for several moments before the overacted sighs, breathing, gasps, eating noises, etc. kick in--that's when it's actually done loading.)
I've just started hitting F1 every time I start a new time period and waiting for the menu to pop up, once it does it's done actively loading all the assets at once. Popping the F1 menu open seems to make the game put all the secondary thinking on pause, animation choosing, AI pathing, etc. so it's not wasting time trying to process all that alongside processing the incoming assets.