This may be the most expensive part of your Tesla coil system. You will need an AC transformer of at least 3,000 volts at 30 milliamps to get the spark gap to fire, and more is better--within reason. Once you exceed 15,000 volts at 60 mA, you're leaving the indoor, tabletop arena and graduating to garage maniac status. This system is designed around a 7,500 volt, 30 mA neon sign transformer.
Until a few years ago, most neon signs were powered by heavy transformers potted in insulating tar. Nowadays concerns for safety (and liability) have made the old heavyweight NST obsolete. Modern neon power supplies are light, solid state units with Ground Fault Interrupt (GFI) protection. This is all very well if you're trying to light up a Michelob sign, but GFI power supplies won't work in a Tesla coil system. GFI is designed to stop just the sort of continual sparking a Tesla coil requires, so avoid modern neon power supplies. If you can't tell by the label if a transformer has GFI, try this rule of thumb: if the unit is small and light, it has GFI and won't work on your Tesla coil. If it's heavy as lead and almost as big as a cinder block, it will.
Other power supplies can be used. Oil burner ignition transformers (OBITS) can often be had cheaply. They almost always come rated at 10,000 volts at 23 mA. As with neon sign transformers, the newest OBITs now come with solid state circuitry and GFI. Pay close attention to make sure you don't get stuck with a useless GFI model.
Some coilers use microwave oven transformers. My advice is, DON'T. MOTs, as they are called, are rude, crude, and develop lethal amounts of current. If you don't know what you're doing, you can easily harm yourself or others. Until you've learned a lot more about electrical engineering, leave MOTs alone.
Car ignition coils are sometimes used to power Tesla coils. Ignition coils have to be driven by some kind of circuitry, either a 555 timer and associated parts, or even by something as simple as a household dimmer switch. This can be a fun project
on its own.
Tesla coils can be powered by DC, but that's a different topic not covered here.
For best and simplest results, get your hands on a neon sign transformer (NST). All the ones I own I got off eBay pretty cheaply. They're heavy and can be expensive to ship, so if your budget is really tight, try a local sign shop. Many businesses are still switching over to electronic GFI power supplies and discarding their old NSTs. You might get a bargain if you offer to take an old-style NST off their hands.
Many NSTs come without power cords. When I salvage parts from old microwave ovens, one thing I always keep is the sturdy, three-wire oven power cord. These are easily adapted to NSTs.
One thing to remember about using NSTs: never move or lift an NST by the ceramic terminals! They're brittle, and they can crack or snap right off, then you've got nothing but a doorstop. Always lift NSTs by the case. One thing you can do to improve the portability of an NST is mount it on a base. I put my 7,500 volt Allanson NST on a shellacked pine plaque, fitted with four rubber appliance 'feet.' This insulates the NST from the tabletop, and gives me a handy place to grab and lift it.
It happens my Allanson unit has a pull-chain on-off switch. This is unusual. Most NSTs I've seen do not have on-off switches. Once you have grafted on a power cord, the best way to handle the NST is through a variac (a variable transformer), or an isolation transformer (see "Variacs and Vacuum Cleaners"). Failing those, you may have to use a wall socket controlled by a wall switch, or else put together a switch using a household wall switch mounted in a utility box.