All you need for this is a 'Planetary Approach Suite', all ships have these by default but just make sure. It's found at the bottom of the optional internal modules section in Outfitting.
'Landing' on a surface can either mean docking at a station or just landing on the actual ground. There are many stations on the surface of planets and moons, you can distinguish these stations from space stations by their icon in the Navigation tab of your Target Panel (
+
) - they look like little buildings as opposed to floating or circular icons. To find stations on planets or moons use this page:
https://eddb.io/stationChange the 'Include Planetary' box to 'Yes' and type in the name of your current system in the 'Reference System' box. All stations with a little 'Globe' icon are surface stations. Simply dock at the station like usual to count as one planetary landing.
To find landable bodies without stations you can use this page:
https://eddb.io/bodySimply change the 'Planetary Landing' box to 'Yes' and type in the name of your current system in the 'Reference System' box.
You'll get a list of all the nearest bodies to land on. To land on the surface simply approach the atmosphere head on at a safe speed, get near the ground and deploy your landing gear, you can slowly coast the surface until the ship hologram turns blue indicating a safe landing site, descend slowly until you lock on to the surface, this counts as one planetary landing. To repeat for the greater achievement of landing on 100 you'll have to take off, escape the atmosphere and then re-enter and land again.