I got a Canon 1300D a month or two ago and I've found that I always have to use high ISOs (800 or above) to get decent exposures, otherwise I'm forced to using shutter speeds that are almost too slow for handheld shots (1/50 or slower) or limited to just large apertures.

That's very unlikely unless it's relatively dark where you are. Remember that human vision works better at adapting to low light and this can mislead people into thinking it's bright when it's actually low light for a camera.

The old rule of thumb is Sunny F16.

I'm confused about this because mom used to use ISO 200 film and had no problems like this - she regularly took handheld landscapes with apertures f/11 or smaller and shutter speeds 1/100 or faster.

That would be Sunny F16. But that refers to a very bright sunny day. You don't say where you are, but if you're in the northern hemisphere it's winter and typically overcast all day - certainly not bright sunny days !

I took a photo of the view from a window this morning in Av mode with ISO 400 and an aperture of f/7.1 and the camera chose a shutter speed of 1/100.

That would suggest it was overcast ( unbroken cloud ) - quite normal condition for a norther hemisphere.

Sound quite reasonable for the camera to meter that way.

To me it sounds like you are shooting in a perfectly normal overcast condition and your camera is correctly metering the scene. It's your human expectation that is the problem.

Camera's don't adapt as well as humans to low light.