Just bought a house with a 1/4 acre pond on the property, Its awesome! We love sitting by the pond and watching our goldfish swim around...

yup, previous owner dumped all her koi/gold fish into the pond when they stopped living there full time 2 -3 years ago, there is easily 40 - 50 currently ranging from < 2 inches to easily 6+ inches long.

I haven't figured out how to catch the little buggers yet, but I want to fish my pond and stock it. Any suggestions on how to figure out if anything else is in there, what to do about the stupid goldfish, and what to stock it with?

Will be used completely for recreational fishing, I don't eat fish.

Don't have aeration yet, looking into that. Previous owner had a fountain I think

this forum is awesome, I'll be spending a while just reading, thank you!

A handful of Northern Pike would do most of the work for you, Walleye would be another possible tool for that use. Low numbers, non reproducing tools.
1/4 ac wouldn't be too bad to trap most with baited traps though..

You essentially have carp in the pond now. Best bet is to kill it off and start over. Others can best suggest how to do this best in your situation.

Agreed. Koi are just colored carp.