I'm building a battery box/inverter setup.  I have a 500 watt inverter that came with 12 gauge wire.  Is that sufficient, or should I go thicker?

I also have a FT-2900R ham radio, what gauge should I use for the charging pigtale/ham radio connection?

you need to know the amps.

watts = volts * amps

or (watts / volts) = amps

Then go to a table like this

wire gauge table

At least 10 ga.

500 watts at 12v would be a little over 40 amps... I'd be tempted to put a little more copper in there even for a very short run.  Just doubling up the 12ga would give you about 9ga, or if you could come up with short pieces of 8ga, something like that.  Part will depend on how long the wire run is, if it's just 6 inches you could get by with less than if it's 10 feet.

For the 50 watt mobile you should be good with 12ga... most of the time 14 is adequate but since you're running off of battery you want to minimize voltage drop as much as possible.  If it's a very short run, like a couple feet or less, 14ga should be fine.
Eh, just reading that the 2900 is a 75 watt radio.... would definitely do 12ga if you are planning to run at full power.  10ga wouldn't be a bad idea for a longer run, or paralleled 14ga like the manufacturers commonly use for 100w HF radios.