High/Low pressure systems - what does it mean weather-wise

They always say in the weather something like there's a low pressure system coming in from the east. Is this a good thing or a bad thing? Personally I've always thought myself -

High pressure = good weather warm, sunny, calm
Low Pressure = bad weather, rain, wind etc

Am I correct in assuming this?

Generally that is true.

Low pressure is caused by rising air, and rising air cools and forms cloud and rain.

Sometimes high pressure can cause air to become trapped near the ground, in autumn this causes fogs. and sometimes low flat cloud can form in an high pressure, which can result in days of dreary grey weather. This sometimes surprises weather forecasters, who had predicted sun.

When a high pressure brings sun, this causes surface temperatures rise.
in front of low pressure, warm and moist air is often pulled North, producing humid weather, that is followed by heavy rain as the cold air replaces it. Low pressure rarely produces "enjoyable" warm weather.

High pressure = good weather warm, sunny, calm
Low Pressure = bad weather, rain, wind etc

Am I correct in assuming this? In general, yes you are correct.

As for the temperature of high and low systems, in the summer it's not unusual for temperatures to be higher under the low-pressure system than they will be after the dividing line ("front") passes and high pressure takes over. This is because in the summer, low-pressure winds pull in moist, warm air, but after frontal passage the wind swings around to the north and the air mass dries and cools considerably.

You are entirely correct. Low pressure systems (extratropical cyclones) form as a result of rotation and lifting, bringing upper level divergence, etc. The end result is rising air, which promotes cloud and storm growth. High pressure systems (anticyclones), form as a result of sinking air, which inhibits cloud and storm growth.