Here are a few general/Geoscape tricks I know.
When you first start the game, the Microprose name flashes up and then the trailer starts. To bypass these screens completely, simply press"X". That brings you directly to the starting screen.
If you accidentally start a new game with the hotspot interface selected, you'll see the globe after selecting the placement of your base. But if you want to get to the menu options to change to the point and click interface, press "select" twice. That should highlight the INTERCEPT menu option in light lavender. Now scroll down to OPTIONS, and do the switch. That's a whole lot easier than reloading the game!
If you have the point and click interface selected, and want to rotate the globe manually without mucking around with the globe control icons in the lower right, do this: First, place the yellow pointer to the place on the globe you want to view. Next, click the "Circle" button. That should rotate the globe to the desired location. If the location you want to see is on the other side of the earth, place the pointer near the very edge of the globe and press the "Circle" button a few times. In order to zoom in and out you still need to use those icons in the lower right.
And yes, the Playstation gets kinda laggy, especially if the game is deciding where to place UFO's and close to the end of the month. Get a year into the game, and the Geoscape time goes by real choppy. You might want to run the time in lower increments (like hours instead of days) to keep the game from going too slow. If you accidentally get caught into a freeze up when the 1 Day time is selected, press the "Circle" button, and the time should slow down momentarily to 1 Hour. Pressing the "Circle" button when 1 hour is selected will reduce the increments to 30 mins. Each press of that button will make the clock move ahead by 1/2 of the current time increment selected. This sometimes helps the game to "decide" what to do next, without wasting an entire afternoon waiting for the Geoscape to go by 1 Day.
I'll have to scrounge up my tricks for the Battlescape for you. Hope that helps for now. If you have any other questions, please ask!