What is the best way to connect a farm fence to a vinyl fence?
The best way is to put two posts directly adjacent to each other - the vinyl one to the front and the wooden one just behind it, or just inside it depending how you want the front corner to look - the wooden one should also be braced to the next wooden one as it's an end-post for a tensioned fence. Putting the wood one inside will probably look better from the road, but mock it up and see what the arbiter of appearance in your household thinks.

Might also matter what critters you are trying to keep in, or out.