Malc asked about first steps in acquiring the skills to be independent. First steps, always, are to make a list of what those skills are. What do you need to be independent? If you are really interested in this, stop reading now, make your own list, then come back. What follows is my 'first thoughts' list.
The big four for survival are shelter, water, fire and food. They are in that order for survival because exposure, generally, will kill you quickest and preserving heat is more efficient in the short term than manufacturing heat. Thirst kills you second quickest. Fire is a tool and, among other things, can make water and food safer. (There is a debate in the survival community about whether fire is more important than water since unboiled water may not be safe). Most of us can go much longer without food than we realize.
These four have very different levels for different situations and time scales. Pure wilderness survival, dropped into a pristine wilderness with few or no tools requires one set of skills, and unless you have a time machine you will really have to work to find that situation. Surviving if your car goes off a road in a blizzard or if the power and water gets shut off to your apartment (blackout and riots, yay!) are different. Weathering a storm for a few days is different than trying to recreate civilization from scratch (time machine or portal, again).
The principles of shelter, water, fire and food (SWFF) are universal, but sometimes it can be hard to find someone who can teach the principles without their perspective creating blindspots (e.g. so into nature that they won't use litter for natural tools or so into he-man survival that they don't admit sometimes you wait for help.)
Aside-- One of the most important exercises is to live at this subsistence level for a time. For however long it takes you to be confident you could last forever. Then you realize how little you really need, and the people (I'm thinking advertisers, but also peer group) who make a living from creating hungers lose power.
And time frame-- wildcrafting food and medicine can take days, but growing it can take months. A garden, even a little one, eases you dependency. If it's more than a few days, waste management becomes a critical skill as well.
So, shelter, water, fire, food. And waste management.
I'd add medicine and defense as critical skills. Readers of this blog probably have their own ideas of defense, but I'll add this: For any likely disaster (say you live in an earthquake or tsunami zone) you should have a 'defend in place' plan and a GOOD (Get Out of Dodge) plan. The GOOD plan must include where you are going. Never run away, always run towards. And you must have a plan (and skills) for defending in place and a plan (and skills) for defending on the move.
For medicine, advanced first aid is a minimum. Go for EMT. There are some excellent home health and medicine books, including Werner's "Where There is no Doctor" and Sehnert's "How to be your own Doctor (Sometimes)". There are limits-- you won't handle a burst appendix by yourself-- but independence like most things is a path more than a destination. It's percentage points.
At a more generic level-- literacy. Including scientific literacy and forensic debate. If you want to know how you are being manipulated you have to understand what science _is_ (the scientific method, not just technology) how statistics works, and the common logical fallacies.
Statistics and trig are, IMO, the two most useful maths. Some geometry.
Critical thinking is huge, but like breathing and walking, everyone thinks they're already natural masters and most can't be objective for shit.
With more room and time, I'd love to matrix this out. For instance, take shelter. At the basic level of skill, it's building a debris hut or burrow. At the basic level of understanding it's knowing that if power is lost you move your whole family to one small room so that the body heat will keep the living space comfortably warm. At the journeyman level, you are learning to repair or build all the things that make a modern house and at the master level, you can build your own home to your own specs...but can be happy living in a debris hut.
For survival skills I've played with Tom Brown's School and the Maine Primitive Skills School. Good skills, but they definitely come with a philosophy that may not be your cup of tea.
My favorite is Toby Cowern. He's smart, he teaches you how to think instead of telling you what to do. His skills cover wilderness (which he practices north of the Arctic Circle) to urban, disaster and even some combat. He only gets to the United States about twice a year, but he's experimenting with on-line and video courses. http://www.treadlightlytraining.com/
Local colleges will have EMT training and there is always the Red Cross. If you're rural, you might be able to volunteer for a local Fire Department and get some good training and experience. Not just in fire suppression and First Aid-- the ICS (Incident Command System, how I was taught to plan operations) was pioneered by FDs.
There are more resources out there than ever before. One more. FEMA has created Community Emergency Response Teams and I hear the training is good:
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4 days ago