Do you know what species you're dealing with? I've done a similar thing with the common British ant, and the Queen was very easy to find.
I know a good species to work with is Leptothorax, cos it's small and lives in flat nests - a good study insect.
Who gets to mate with the Queen depends on the species - but essentially what you need is her, and some worker to provide the sperm, or get larvae if you can...
edit to add: to find the Queen - simply follow the workers, when you see them, they are foraging for food to take back to the nest. She stays put in the centre for most of the time, and larvae will be found in her vicinity.
Usually the queen is predetermined from birth, so no, you can't start a colony with just a worker ant. They are all haploid (half the genes) and sterile. It's better to think of the colony as the organism and the ants as specialised cells.
eg The workers are the differentiated cells, like skin, organs etc The Queen is analogous to the somatic cells and so reproduction occurs via her and select males only.
All depends on species too.