My ac unit went out last week. I got the meter out and found that there was no 24v coming out of the transformer. I checked to make sure there was voltage coming in, and there is. I replaced the transformer with a new one, made sure it was hooked up properly, and still no 24v. Also... where the main contact switch is, the contact is open, and when I check the voltage across, it is if the contact switch is closed. Nowhere on this machine is there a name or model number. There is no safety switch for the panel.

If you have continuity on the primary and secondary windings, it should work.
You need to see if you have 24 volts out, while having 240 volts to primary.
If it burnt out the transformer, add a fuse to protect it and see if you have a short on your low voltage control wiring( To thermostat and to outside compressor), or shorted windings on the outside compressor contactors coil or orther relays/contactors.

Another thought, if it has a float switch and is up(has water), it will interrupt power to the thermostat, If UV, should have a door switch.

It doesn't have a float switch, or a door switch. The transformer has 240 going in but no 24 coming out. It is a brand new transformer.

A short will destroy a New transformer. If you have voltage to the primary, and none on the secondary, it IS Shot.
Reread my first post.
Good Luck.
Next time, Try just the fan, then heat with fan.
Then try cool, if if fuse blows when set to cool, wiring to compressor contactor, or contactor coil is shorted.
When it blows, it may tell which relay/contactor coil or associated wiring is shorted.

A short will destroy a New transformer. If you have voltage to the primary, and none on the secondary, it IS Shot.
Reread my first post.
Good Luck.
Next time, Try just the fan, then heat with fan.
Then try cool, if if fuse blows when set to cool, wiring to compressor contactor, or contactor coil is shorted.
When it blows, it may tell which relay/contactor coil or asociated wiring is shorted.

How this all started... my blower motor quit working and the coils iced over. I turned it off, let it thaw and tried to restart it. No blower motor. Got to checking and noticed a wire terminal was old and loose on the blower motor. I replaced the terminal. I tried to fire it up and it ran fine. I turned off the breaker, and I put all the panels back on. I tried to fire it up after that and nothing. The contactor wouldn't even click. So I pulled everything back off and went around with my meter and found I wasn't getting 24 volts. I replaced the transformer with a new one, still the same. The old transformer was definitely bad. It had the death smell. Even though my contactor is open, there is power on both sides of it, as if it were closed. If I manually press the contactor, still nothing.

A transformer is basically 2 coils of wire. The Primary winding, and the secondary winding. To Transform from 240 volts to 24 volts, You need 10 times as many windings.
Each coil will have continuity if it is not burnt open.
There is no internal switch, Next time try hooking up just the primary with the secondary windings clear, apply power and check for you 24 volts out.
Hook up 1 secondary wire and see if you have 24 volts still, then hook up second wire, if it blows, transformer was working and there is still a short.
With secondary wires disconnected, verify you have no continuity between them and ground.
Add A Fuse, unless you can get transformers cheaper than fuses.
I also wonder if the wiring is getting pinched from the cover as you touched nothing else?