What are the first things you should do when starting with a minimalistic build (1 anvil, 1 copper nugget) in Dwarf Fortress.
A true minimalistic build is 1 anvil and 2 copper nuggets; so I'm going to assume that's what you meant.
This gets you a forge, a smelter, an axe and a pick, thus completing the minimalist challenge.
Or you could just start out with 2 ash, 2 coal, an axe, a pick and an anvil and skip this otherwise tedious beginning.
Its worth noting that with 1 copper nugget you have the choice of either a pick or an axe. As a result you either don't have enough copper or not enough wood to complete the challenge.
Edit: actually if you had above ground lava, you could avoid needing the second copper nugget and make two ash giving you have enough for the forge (as lava forges don't need coals).
In DF2010 (the latest build) you can use the training axe in place of a normal axe to cut down more trees. People who traditionally used the minimalistic build consider this "cheating" as you essentially get your axe for free. This can let you use 1 copper nugget.
This gets you your forge, smelter, axe and pick. Everything you need to build a full society.
A true minimalistic build is nothing but a wagon; but I'm going to assume that's not what you meant.
The idea is to survive the summer until the caravan arrives in the fall. Before the introduction of training axes, this meant you had only three logs with which you would need to use sparingly until fall when you built a trade depot from them. You also need to build up enough wealth to purchase some tools in the fall, which you can do by gathering, then farming those seeds, fishing, and cooking the food you have into prepared meals. Now that training axes can cut down trees, you can use one of your three logs from your wagon to make an axe, then build a wooden fortress, barrels and a still, etc.
If you try this out, good luck! I myself usually spend all my starting points, but bring along an anvil and ore in lieu of the more expensive zero-quality tools.
assign a drinking area, hopefully the water is potable start gathering plants, easiest source of food, and will provide seeds when the dorfs eat dig for more metal, your going to want another pick and a couple axes
lastly, build things, your minimalistic build will greatly benefit from the incoming waves of migrants. Masons using the rock you dig up can up your fortress value attracting more migrants