So last year, I had my AC recharged twice. Both times the air was ice cold for 2-3 days and it all leaked out. So, I know that my system works, but all the refrigerant is escaping somewhere.

Can I figure out where it is leaking out on my own? Or do I need some access to some special tools I probably won't have...

Well, if it is a big enough leak, you can find it with bubble soap. If not, a leak detector is what you'll need. Also, possibly look for oil stains, or dirty spots anywhere on the AC system. The oil leaks out with the refridgerant, makes a stain, collects dust. But, you can't readily see what'* going on with the evaporator. A little seepage around fittings is normal.
Also, keep in mind that you are losing oil too. Just recharging it will not replace that oil.

Usually the ac compressor leaks the most, specially if its the stock one, since its the only part that operates the whole system. check on the bottom of the compressor for massive dirt build up.

And I've noticed leaks from the Schrader Valves, a primary source.
Can be secured with your standard bike tire tool.