Rain, yes. Snow, yes. Wind, yes. Tornadoes, yes.
General weather phenomenons that can happen on Earth happen on other planets of our solar system, as well as any Earth-like planet outside our solar system. Depending on the planet, it may also have things that don't really happen on Earth. For example, on Mars, clouds are made of ice due to the lower temperature. On Earth this only happens at the polar regions, where it is so cold that ice clouds can form. Also Mars has planetary dust storms that can obscure the surface of most of the planet for months at a time. If your Earth-like planet doesn't have lots of rooted organisms or has lots of deserts, this may be a weather effect that you can employ. Furthermore, Mars has very large mountains, and they are so large that they block lots of weather. I believe the top of Olympus Mons has 1/3 the atmosphere than the surface.
Keep in mind that planet size has an effect on wind patterns. Earth is the largest terrestrial planet in our solar system and has three cells per hemisphere, but other planets have different cells.
Earth is rather straightforward, with 3 cells per hemisphere. Due to the less dense Martian atmosphere, during the summer it will have 2 cells per hemisphere, and in the summer (shown), it will only have 2 polar cells and a central cell that crosses the equator. Venus has a more complex system, since its atmosphere is very very dense. Closer to the surface it has two hadley cells and a polar double vortex, shown here:
An artist's rendition is shown here:
Pretty weird! Also, since Venus has a very thick atmosphere and a very slow rotation time, some parts of the planet will be under direct sunlight for many days at a time. This creates a subsolar cell that moves with the part of the planet that receiving the Sun's direct sunlight at a particular time. The night side of the planet does not have this, but after a few Earth months, the planet will have rotated such that the night side is now the day side, and now the day side will have that weather pattern.
If you are planning an Earth planet with similar atmospheric density, similar size, and similar rotation time, then you shouldn't have to worry about this, but if you make any changes look at what we have seen in the Venusian and Martian atmospheres.