How do I drill holes in pottery?

Carefully. Use a masonry bit in a cordless drill & s l o w l y let the drill bite into the ceramic.

Use a glass or tile drill bit.

https://www.lowes.com/pd/Bosch-1-4-in-x-2-in-High-speed-Steel-Flat-Rotary-Drill-Masonry-Drill-Bit-for-Tile/1000035017

Ed’s link to the proper bit is correct but keeping cold water inmthe depression between the foot ring of the pot and the hole you are drilling will keep the bit from wearing out before you get the two holes drilled. Hold the drill/bit on an angle to start so that you create a sort of pilot depression before holding the drill horizontally to drill the hole.

Added to these answers use tape to drill into it steadied the drill and gives a clean result. Position tape of any sort really apart from cellotape b4 drilling

Above answers are all excellent but I also made a "well" of modeling clay rolled into a bead around the hole to hold the water.

I use my dremel cordless and a diamond tip bit for cutting holes in glass, it cuts porcelain and pottery fine

Use an appropriately sized carbide spade drill bit (glass, porcelain and tile) and maintain the recommended speed for the size of the bit while drilling. Put an inch or so of cold water in the pot and add a squirt of dishwashing detergent then mix. Put some more water and detergent in a squirt bottle of some sort. Use a double insulated corded drill motor else a battery powered motor. Maintain just enough pressure on the bit at speed to cut through the glaze and clay core of the piece until the bit just begins to break through the other side. Then turn the pot over and slowly finish the hole while cooling/lubricating the bit with a stream from the squirt bottle. Use safety glasses at all times during the drilling process. If you have a aluminum oxide or garnet point mounted on a mandrel, you can use it (wet or dry) in your drill motor to smooth the sharp edges left by the drill bit; just don't overheat the pot material while doing so. Otherwise, roll a strip of aluminum oxide or wet-or-dry sandpaper into a cigarette shape and sand the edges smooth. Mind your finger tips though, the edges may be glass shard sharp.

Use a diamond bit. I use a cordless drill and let cool water run over the bit.