First of all you need to decide exactly what 'self sufficient' means, the most rigorous definition would be that once they arrive they can only use local resources and must trade for anything else they need.
This depends a bit of climate and terrain. Clearly the ideal situation is to have a favourable climate and plenty of flat fertile land. You would certainly want a variety of crops and livestock both to mitigate any failures of one particular crop and to ensure a reasonably balanced diet. On an island fishing is a good way to supplement the diet it may also be worth creating inland ponds to farm fresh water fish.
As well as food agriculture can provide a lot of other useful resources. You will want fibres for textiles and cordage, plant and animal fats can be used for fuel, lighting, lubrication and waterproofing and leather from hides is a very versatile material. Also fast growing woods such as willow can be coppiced to provide both fuel and building material.
Careful choice of crops will help here, although bulk crops like wheat and an efficient way to get the most calories out of a given area of land this type of farming is very labour intensive without mechanisation and for a small population a mix of livestock and a kitchen garden/smallholding approach may be more manageable.
You may also want to manage more mature woodland to get the most out of it as it can provide a variety of useful wild plants, a habitat for game as well as timber for fuel, tools and building. Indeed if you have a big wood and a small population it may be a more labour efficient resource than farming.
This should be fairly straightforward, there are building techniques to suit any range of local materials and conditions you can think of although you may require some specialist skills to make the best use of them for example a round house design is a good way to build with natural materials and simple tools.
There are some basic tools that you will need to be able to make and maintain for any reasonable definition of self sufficiency, spades, knives, axes carpentry tools and cooking utensils will all be vital in the long term. A properly trained blacksmith should be able to provide all of this with the main energy requirement being charcoal to fuel the forge. A water wheel would also be very useful as it can provide motive power for bellows, grinders, drop hammers etc which will greatly reduce the labour required.
So a forge will allow you to keep all of your other trades supplied with well maintained tools and equipment. Unless you are lucky enough to have easily accessible iron deposits on your island a good stock of steel and other metals will be vital as this is one thing that you can't get from farming. On the plus side most of the metals you will be using are readily recyclable.
A basic engineering workshop with welding and machining facilities would be very valuable and this would probably have first call on any source of electricity you have.
Clothing and textiles
Making clothing from scratch is pretty labour intensive but it is also an activity which lends itself well to doing during downtime in evenings and winters where there isn't much else to do and at least isn't as physically hard as digging or chopping wood. The main materials will be plant and animal fibres and leather.
This is a bit of a stumbling block as you will never have full access to modern medical facilities and be truly self sufficient. It is also expensive whether you pay for it directly on demand, through insurance or indirectly via taxation. So you need to decide what you would do in a real medical emergency.
Anything like a modern electricity supply probably won't be sustainable so you will certainly need to prioritise what you really need. Bear in mind also that a lot of your time will be taken up with essential tasks so sitting watching tv is probably not on the agenda. Heating lighting and cooking can all be taken care of by wood, charcoal, candles, oil lamps etc and your biggest priority for any electricity supply you do have would be functional tools.
There is also the problem that once any electronic equipment breaks down you likely have no way to repair it and really what electronics will you need ?
A much higher priority than the internet would be a decent library with a good selection of reference books on agriculture, animal husbandry, horticulture, medicine, woodwork, engineering etc etc.
For power a good bet would be a small steam engine, these can use any solid fuel and are realistic to maintain long term with basic technology. As well as running a generator they can be used to give direct motive power for driving various machinery.
The trouble with solar panels is that as well as being intermittent they have a limited life and once they fail you have no way to replace them.
The question mentions buying luxuries and this raises the question of money. If you really are self sufficient then that suggests that you are able to trade any surplus goods you produce. The problem here is that your have a adopted a very inefficient lifestyle from a modern commercial perspective and the chances are that a lot of your labour is going into basic subsistence so even in the best case scenario cash is going to be very limited and 'luxury items' will take on a rather different meaning.
With this in mind any cash you do have would be reserved for things which you absolutely can't produce yourself and are of real benefit.