What can I do about water coming in from underneath when it rains?
I would try to identify the source of the water.

You don't have gutters. If rainwater runs off your roof and splashes onto the ground next to your walls I would consider installing guttering and leading the rainwater through pipework away from the house to a soakaway (large gravel filled pit).

If the surrounding ground slopes towards the house, I would consider installing something like a French-drain that intercepts ground water and directs it around the house to a point where the ground slopes away from the house. This is a big project and you need to be sure of what  is causing your problem before starting.

In both cases you probably need professional help or at least get someone experienced to come and look at the building.