Partly what your colony needs will be determined based upon the specifics of what you mean by self-sufficient.
Do you mean "economically" or do you mean "catastrophe strikes the Earth and the colony needs to survive on its own?"
Regardless of your definitions the initial requirements will be similar for either type (and those are very well defined by other answers). Basically figure out high mass needs, especially those that do not require special processing, and fill those needs first.
As mentioned in other answers, these will start as things like:
- Breathable air
- Potable water
- Productive Soil
- Radiation shielding
- Construction materials
The hurdle of providing these things to your new colony is a lot lower if you do a good job of picking a location in which many of these things are readily available without a great deal of processing - and it is why finding water at the Moon's poles is so exciting.
Assuming you want economic self-sufficiency, as the colony becomes self-sufficient in these high-mass colony needs, the colony needs to determine what valuable low-mass things and services it can produce to compete with those provided by Earth.
Initially, these might be:
- Servicing satellites
- Refueling of satellites
- Clearing space junk out of Earth orbit.
As the colony continues to grow and establishes production infrastructure the colony will begin exploring goods and services it can create that are unique or superior to those available on Earth.
These might be:
- Astronomical facilities and research facilities (no atmosphere, longer observation baseline)
- Other scientific facilities and research (novel environments)
- Biomedical research (novel environments)
- Drug research (zero gravity is good for growing large protein
- Specialty Electronics (zero gravity and high vacuum -> extra pure materials)
- Specialized high value mechanical items (e.g. zero-gravity ->
perfect ball bearings)
- Novelty items (Moon rocks, etc. for collectors)
- Specialized materials (zero-gravity & high vacuum makes especially
large and perfect crystal growth possible - or perhaps we'll be able
to make true high-temperature superconductors or super-long
nanotubes - km instead of mm in length)
Ultimately, economic self-sufficiency won't be enough for a colony. It'll need to be able to survive without resupply (if necessary).
IMO, this is where 3D printing (and other "additive manufacturing" techniques) are a really exciting development. If you can supply the raw materials many different types items can be created with them as long as you have the design program.
Need a locking mechanism for your airlock? Program the part into your 3D printer, fill the feed bin with the proper materials, and a few hours later, voila: a new airlock mechanism.
Such devices will never replace high volume (& therefore cheap & high speed) manufacturing methods but would be perfect for the needs of a space colony.
So the real requirement would be to have the ability to build those 3D printer devices.
I view a population like a pyramid. You need a certain number of people at each level to support developments at higher levels. A space colony (with our current levels of knowledge and technology) will require lots of specialized knowledge and skills to keep going.
In a terrestrial country, it requires a certain population base to maintain a high technology levels. Some city-states side-step the issue by being embedded in a larger country that provides that support base.
But what that means is that the colony will either need to support a huge population, or a lot of work and technological advances will need to be achieved to "dumb down" the knowledge and skill requirements of the colony. So instead of an active closed-loop environmental system that requires human maintenance, we'd have research and develop a passive environmental system that maintains itself (think spaceship Earth).
Except for perishable items, transportation will use the Interplanetary Transportation Network. Such trajectories use much less energy ($\Delta V$) than low-energy Hohmann Orbit transfers but can take decades to complete.
Humans or some other perishable items will at least require Hohmann Orbit Transfers.