From the earliest times that people first laid seed in the ground, farmers have watched the weather, trying to figure out what it's going to do next. It all started with trying to plant so that the crops aren't going to be killed by a late or early frost, or too much summer heat, or too much or too little water. Today, everyone's still trying to guess what the weather's going to be like tomorrow, only now it's mostly for planning commutes around it and hoping weekends don't get rained out.
The basic indicators of all weather are temperature, pressure, humidity, wind direction, and the types of clouds in the sky. Whenever the pressure goes down, rain or snow is on the way. The same thing goes for anytime the temperature starts going up or down beyond normal day-night changes, which indicates a front that's going through. In the summer, when it's humid enough, even normal daytime heating can set off a thunderstorm. Watch for fluffy cumulus clouds which start to grow huge, with dark bases.
A sure sign of coming bad weather in the summer is the high wispy clouds known as mares' tails, especially if they're accompanied by a stiff east wind. These cirrus clouds are the earliest indicators that there's a cold front on the way, which usually means severe thunderstorms if you've got enough humidity in the atmosphere. If you keep watching, you'll notice that it looks like the cirrus clouds start clouding over the whole sky. What's really happening is that the stratospheric cirrus clouds are being replaced by low-hanging cumulonimbus clouds. Some cumulonimbus clouds can reach up all the way to the stratosphere. Those ones cause really severe storms and even tornadoes.
Wind direction is caused by the interaction of high and low pressure areas. Winds spin around a high pressure area clockwise and a low pressure area counterclockwise. (It's the other way around in the Southern Hemisphere.) An approaching low pressure area is usually accompanied by a stiff east wind. Nor'easters are named for their strong northeast winds. West and northwest winds usually mean clear skies and good weather, unless it's winter and you live in the snow belt just east of the Great Lakes. Then it means that you're about to get hammered by lake effect snow.
For a rainbow, you're always looking into the rain area with the sun behind you. Rainbows to the east mean the storm is past. They can only happen in the late afternoon or evening. Rainbows to the west mean the rain is coming. They can only happen in the morning.
Halos around the sun or moon mean that there's ice crystals high in the atmosphere. Moon halos usually mean rain on the way, same as cirrus clouds do. Sun halos usually happen only when it's bitter cold, too cold for most heavy snow, so they usually mean blue skies and cold weather.
That's about as much as you can get just by looking up at the sky. For more than that, you've got to get more information. In Canada, weather travels more or less west to east, so you can guess what's coming by looking west or by watching what your neighbour to the west is getting. If the clouds are to your east, most of the time the storm's already past.
Satellite pictures show the large-scale movement of clouds and whole weather systems. That tells you the direction of the jet stream. If you know which way the jet stream is looping, you can predict the weather by looking "upwind" along the jet stream. Unstable systems which cross open water usually pick up a lot of moisture from that water and become major rain and snow events, especially if the water is warm. If they don't cross open water and there's not a lot of humidity in the air, they bring lots of wind but not a lot of precipitation.
For more than that, you pretty much have to be a trained meteorologist. Meteorologists have degrees in the physics of the atmosphere, which they use to figure out how weather systems are going to interact with the local geography and with each other. That gets very complicated, but all that is reduced to a simple percentage chance of rain on your weekly weather forecast. There's so many factors that go into it, it's amazing they get so much of it right.
Or you could always just rely on your trick knee. Aching bones and headaches are often a sign of bad weather on the way.