The ADS works the same as the cheaper discovery scanners. It's benefit is that it has no maximum discovery range within a system.
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Further to the other answers, it is possible to explore with the cheapest discovery scanner - however there's a significant risk of failing to spot celestial bodies if they are very far out.
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Anything that doesn't show up on arrival at a star and the first use of the (cheap) discovery scanner can only really be found by eye, using parallax shift. The backdrop of stars doesn't move relative to your position. When in fast forward FSD mode, any dots that appear to be moving against this backdrop are celestial objects. Get close enough to them, and do another scan. Make sure you have orbit lines turned on, go to the furthest object, and do a circuit of its orbit looking outward for more celestial bodies "moving" against the backdrop. Rinse and repeat.
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When you think there are no more orbiting planets, look at the numbers for the ones you've explored. Any that are missing from the sequence still need to be discovered, but you know where their orbits are as they are numbered in ascending order out from the star. Just find the orbits of the two wither side, and you'll find your missing planet just by flying above or below the orbital plane and looking at the expected orbital track for a moving dot.
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Watching the route of NPC ships may also help locate planets, as will a look at the nav panel on arrival to see if there are any stations - they have to orbit something (sometimes it's the star, though).
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Finally, look at the star - if it has its own orbit line, then there are other stars or similar bodies to be discovered. They might be very,
very far away, though.
