My a/c went out this morning and will do absolutely nothing. I try and turn it on from the stat and nothing happens. I checked both of the breakers and they are good. Could I need a new thermostat or could it be something more sinister?

Could be as simple as 240 volt to 24 volt transformer, or transformer fuse inside Air Handler Unit. Turn power off while looking inside. Your 24 volts is how thermostat
Sends 24 volts to relays and contactors to operate, Fan, Compressor, Heat Strip,.
Don't Panic, may be easy AND cheap repair

Aha! Now we get to my ignorance of all things. The Air Handler Unit is the inside unit? And where would these fuses/transformers be found on the unit?

Should be a disconnect next to unit, Turn Off, follow the Wiring from Disconnect into unit,
Usually one side or the other.
For front panel, may have to remove 2 or 4 screws. Look for a transformer, 240 volts ac going in, 24 volts ac coming out, and follow to possible fuse. Should have 24 volts after fuse.

Okay, so I think I found the fuse (it is red and as far as I could see the only one in there, but it looks like it is connected to the end of the wire) two questions, how do I tell if it is blown and how do I take it off/replace it?

If you have a fuse then it will look like an automotive type fuse plugged into a printed circuit board. Is this a furnace or air handler ? Old or new?

A fuse will have 2 wires on it. Measure continuity across fuse. Or If Capable, with power ON, measure voltage out from transformer and from fuse. If you have no tester, try push in and turn to remove, replace. It will or will not work. I believe this may be your problem.
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You could have someone near outside unit, and you switch from off to cool, and back, to see if you can hear a click.(Contactor(relay) Energizing and deenergizing) If you DO, your transformer is OK. I don't think it will. Still think 24 volt thermostat control problem
Transformer or fuse.

I didn't see any circuit board-like elements in there, and nothing that looks like an automotive fuse.. is it possible it doesn't have one?

Yes ,not all have these fuses. The first thing you need to check is that you have power to your air handler or furnace which ever you have , if you do then we move on to the transformer, ( this is a square object 2"X2" with wires attached) probably be warm to the touch. The transformer supplys control voltage to operate (24volts) everything, if it goes bad nothing works, is it humming ? Best to check with a volt meter.
What type system do you have ? Gas furnace or heat pump?

Since I know for sure I don't have a gas furnace I assume I have a heat pump. The transformer I found, it is not humming.

Or you also shut your power off at breakers turn your thermostat to off and take the cover off exposing the wires take the red and the green tape them together and restore power then turn your unit to fan on if your inside fan comes on you know it is not your thermostat. Before you get into the transformer unless you do have a voltmeter.

Without a volt meter you are quessing as to what is wrong , if it is the transformer then you may have another problem that caused it to go bad such as shorted thermostat wires or a faulty control componet such as the contactor ,relay, solenoid coil.
If you replace the transformer I would suggest adding an inline fuse to protect it.

My thermostat has blue yellow white and pink wires

If you touch the Pink(May used to be red)to yellow or blue, fan or compressor should kick on. White may be for heat. Colors could be anything, I would try it.
This bypasses thermostat. If it works, thermostat. If battery thermostat, replace battery.