Planets are discovered in many ways, and it has changed over time. The inner planets of our solar system were discovered by looking at the skies, and some observers noticed that some of the "stars" wandered in the sky relative to the other ones. Mercury, Mars, Venus, Jupiter, and Saturn have all been known for as long as we know, as these can easily be seen with the naked eye.

The outer planets in our solar system, Uranus and Neptune, were discovered by telescope. Uranus can be seen with the naked eye, but it's very dim, so it was discovered by a telescope. Neptune was also discovered with a telescope, but it's so dim that you can't see it with the naked eye.

In all these cases, we look at sunlight that is reflected off the planet. However, more recently, we have been discovering "exoplanets," which are planets in other solar systems. In these cases, the brightest thing in these alien solar systems is the sun, so that's all we can see. The way we "see" exoplanets is actually indirectly by looking at how the brightness of the sun changes. When a planet passes in between us and the sun we're looking at, it blocks a little of the sunlight. If we look at the sun for long enough, we can see that sunlight gets blocked in regular intervals, and these correspond to the planets orbiting the sun. Bigger planets block more light, so we can even determine the rough size.

That depends on the planet.

Mercury, Venus, Mars, Jupiter, and Saturn are all visible in the night sky without a telescope, and so their existence has been known for thousands of years, as long as humans have been making careful observations of the night sky.

Ceres, Vesta, Uranus, Neptune, Pluto, and Eris were all discovered using telescopes. It is possible to see any of these (dwarf) planets using a telescope if you know where to look. Since most of the planets in our solar system move in the same plane as the other planets, scanning the plane of the solar system (which passes through the Zodiac, which is why we call it the Zodiac) will help you find most of these planets.

There is a supposed major planet that is outside of the orbits of Neptune and Pluto that has been speculated on based on the gravity it exerts on the orbits of Neptune, Pluto, and other objects out at that distance from the sun. We haven't detected it directly yet, and I am not sure how certain astronomers are that it exists yet.

Planets orbiting other stars are detected in one of two ways: either by observing the gravitational influence of the planet orbiting on its star (possible for large planets but not small ones), or by observing drops in the star's light output when the planet "occults" (i.e. partially eclipses) the star. We do not yet have telescopes powerful enough to spot these planets directly.