What sort of sander do I need to buy?

A palm or detail sander is small so can be used in corners and for small parts, it usually uses triangular sandpaper on a vibrating pad.

A sheet sander uses a rectangular sheet of sandpaper on a vibrating pad, it is similar to a palm sander but for larger areas.

A belt sander uses a loop of sandpaper (the belt) that is rotated around rollers at each end of the machine, it is more powerful, quicker and more effective for large flat surfaces but not much use for smaller areas. You can get machines with different width belts, but 3 inches is quite common.

An orbital sander uses a rotating sandpaper disc, it can leave circular marks on surfaces. A random orbital sander moves the whole sanding disc in another circular motion (eccentric) as well as the main rotation so gives better coverage an fewer problems with surface marking.

Choosing one type would be a big compromise for the variety of areas that you need to sand. I would get a belt sander for the large flat surfaces and a detail or palm sander for the smaller areas. The vibrating palm and sheet sanders can become uncomfortable to use for long periods.
The black and decker one you have seen would be OK for smaller areas but slow for the larger surfaces.

Here are a couple of reasonably priced sanders :
http://www.screwfix.com/prods/74174/Power-Tools/Sanders/Erbauer-ERB907-3-Belt-Sander-230V
http://www.screwfix.com/prods/98623/Power-Tools/Sanders/Ryobi-Corner-Sander

or a budget detail sander :
http://www.screwfix.com/prods/89324/Power-Tools/Sanders/Titan-PGFD280E-Detail-Sander-240V

Depends on the quality of finish you are looking for and the amount of time you are prepared to spend on it. In general you don't need a power sander at all in furniture refinishing. not unless the surface is in poor state of repair. The lazy way is to paint over the varnish after keying it by hand with sandpaper. 120 grit should be sufficient for this. Make sure the varnish is in very good condition. If not it will lift and bring your paint up with it. Best way is to remove it. not with a sander. Use a chemical stripper e.g. Nitromors or Strypit. Clean any residue off and then wash and scrub thoroughly with meths. This neutralises any stripper in the grain and it won't attack your new paint. Neither will it raise the grain. Now sand it. If you must use a power sander then use a random orbital sander. Make sure the machine is running before you put it on the material. Having it on the wood stationary and then starting it can cause it to gouge out wood. Causes more work. This applies to all power sanders. These machines are good for large flat surfaces, drawer fronts etc. Do harder to reach places, mouldings etc by hand. Remove door furniture, knobs, handles etc before sanding. Gives you a clear run and even finish. Don't go higher than 180 grit.

As a rule for the sort of work you are doing I would recommend a detail sander and a random orbital.

Belt sanders are difficult to control and are best at removing large amounts of material quickly with not much regard to the quality of finish.

Orbital sanders are ok but inferior to the random. Orbital sanders leave marks that have to be sanded out, usually by hand.

On skirtings use the large sander for the flat area and the detail sander or sand by hand for the moulding.

Your furniture should be a smooth finish to start with so use anything between 120-180 grit for 'keying the varnish. If the drawers are nice and smooth use 180. Your skirtings will not have a good finish if they are still bare wood so go from 100/120 working up to 180 for a really good finish. If they are painted 'key' the paint with 120.

To paint over the 'keyed' varnished surfaces first wipe it with a clean rag dampened with white spirit. this will remove all traces of dust and won't raise the grain. It will evapourate off in a few minutes, no need to dry it. Apply 2 coats of undercoat and 1 of gloss.

For the bare wood drawers you need to seal the grain. So after wiping with the rag treat with wood primer first. then 2 undercoat/1 gloss as before.

If you ever decide to varnish wood then do it this way. If it is polyurethane based dilute 60/40 with white spirit. i.e. every 100ml = 60ml white spirit to 40ml varnish. It will dry very quickly. Give it a very very very light sanding by hand to de-nib it. Do not use a machine. Next coat is 40/60 white spirit to varnish, slightly thicker. Will dry quickly. De-nib as before. Final coat is pure varnish straight from the tin. You will only need a thin coat. Reason for the two diluted coats is so that the varnish penetrates the wood. If you use it straight out of the tin it just 'sits' on the surface, you won't get a good finish and it is prone to premature peeling and chipping.

Guide to sandpaper grits
100 and below for rough work. Initial sanding of floorboards you might start with 24 or 36 grit. for example.
100-120 final finish for floorboards/stair treads and risers, initial sanding of neglected furniture

120-160 use on furniture to remove previous sanding marks
180-240 final finish for furniture. 240 really only used when the surface is going to be polished.

400 Wet and Dry. Used dry to de-nib varnish etc.

Done exactly that with the walls in my 150 yr old house with the original plaster still on the walls. If the holes are relatively small I prefer Tetrion in powder, not ready mixed. You can mix up just the amount you need and to the consistency you prefer and little waste. Cheap for a quality filler too. Prepare the holes first, brush out loose debris and 'undercut' the bottom of the hole. By this I mean with a screwdriver or similar gently make the bottom of the hole wider than the top. Lightly dampen the hole with a little clean water before applying the filler. When you press in the filler it will form a cone shape and be difficult for it to fall out. Apply filler using a broad flexible scraper or trowel. Overfill it and then using the scraper/trowel with its edge firmly pressed against the surrounding plaster draw it over the hole to finish the filler more or less flush with the existing level. Allow to thoroughly dry. Should be hardly any sanding to do. If holes are quite large you may have to build it up and not try to fill all in one go. If you haven't left too much proud you won't need an electric sander. A block of wood with some sand paper wrapped around it will do and filler is easy to sand. Another reason to steer clear of an electric sander on this type of surface is vibration. You won't believe how disappointed you feel when you are sanding your filler down and see an 18"x12" piece of your old plaster fall on the floor a couple of feet away. Don't forget to size your walls before papering. I usually seal them with PVA mixed with water.

Belt sanders are not really for fine decorating preparation. They are designed for more for heavy duty work e.g floors, stairs, rough sawn timber, removing rust etc. They are difficult to control.

Random orbit is aboout the best you can get for all round finishing (apart from hand sanding) palm and detail sanders are good too, but for smaller surface areas. If you are thinking of sanding large areas of a wall then also think fatigue. Sanders are heavy and get a lot heavier when not resting on a flat horizontal surface.
Wear a dust mask and make sure you have any open cuts covered. If you have old plaster like mine it will be lime based and it can hurt like hell if you get any in a cut.

Looked at your links and personally I wouldn't buy any of them, but then you aren't me.