It's easy to drill holes in clay and ceramic pots if you use the right tools.
Drilling Holes in Pots
The 2 types of drill bits most appropriate for drilling holes in pots are diamond core bits for the highly vitrified (hard material - baked at very high temps - glass, ceramic, ......) pots, and a drill with several names for drilling clay pots fired at low temps, such as terra cotta. That (carbide) drill is called a 3-point drill, spear point drill, or a spade drill, seen here:
For the hard stuff, diamond core drills are best, though the 3 point drill will drill the hard stuff, too.
Diamond core bits ^^^ Start these bits at a 45-60* angle so the bit doesn't wander. Gradually straighten the bit to 90* to the work surface as you drill.
Both types of bits are available in many sizes, with bits sized 3/8 or 1/2" best for pots.
You can buy either type at big box stores, and they're not that expensive, but you won't find a 'set' of diamond core bits worth buying for $9. The bits last a long time, depending on quality and how you use/care for them. They are best used with a steady stream of coolant from a squeeze bottle (contact lens solution bottles work great for this) or with the work surface barely submerged. IE, put the pot upside down in a tub and fill the tub with water until it just covers the work surface. If you use a spritzer or squirt bottle, a 50/50 mix of antifreeze and water is an excellent coolant, and soapy water is better than plain water but not as good as the antifreeze mix. The coolant keeps the tool from over-heating, and in doing so, increases the life of the tool substantially - like triple or quadruple the number of holes you can drill. While drilling, apply very little pressure, just the weight of the drill motor with you providing only guidance is sufficient. Drill at medium speed – somewhere around 1,600 rpm. NEVER use either type of drill mentioned above without a lubricated coolant. The heat will cause thermal shock which will fracture glass and ceramics, and heat build-up will destroy the tool.
Regular masonry bits are "ok" for drilling terra cotta or cement (you can make them work), but inappropriate for anything harder. The right tool makes everything easier.
Masonry bit ^^^
If you have a plastic pot that needs a hole, best to heat up a nail or screw and melt the hole through the bottom of the pot. Drilling many types of plastics leaves microscopic fractures at the hole perimeter that compromises the integrity of the material. Melting the hole seals the edges of the holes and prevents break-outs.
Al