This is a very short post because, these days, metal figures from most quality manufacturers arrive with very little flash or mould marks, and plastic figures (which I rarely come into contact with) only require a wash in detergent to remove the mould release agent. However, figures do require some element of flash to be removed from the bottom of the base, and metal strands* need removing from some extremities and deep undercuts. Occasionally, more work is needed to remove flash. These are the tools I use to remove flash and prepare figures. 
I use a heavy blade, like a Stanley knife, to remove flash from the underside of a figure, and any other heavy flash. I use small pincers to remove 'sprues' connecting one bit of a figure (such as one connecting the bottom of a sword scabbard to the figure's base) before finishing with a scalpel** and files. Scalpels and needle files are also useful for cleaning up other small imperfections. Very rarely, especially where a mould has become worn and damaged, you get a heavier piece, or lump of flash. Where it's possible to reach it, I find a modelling drill the easiest way to remove the bulk of it, drilling it out, sometimes filling in the hole with putty (Milliput or Green Stuff) and reshaping the casting afterwards. Most commonly, I find this happens between a horses legs. Sometimes hands also need drilling to take flag staffs, lances or spears, and although a pin vice will do the job, a modelling drill will do it quicker. When you have one you will use it - so I advise any serious painter / modeller to buy one. I always clean figures with the figure pressed down on a small kitchen cutting board as it allows you to cut down and away from yourself to prevent injury without blunting the blade on any hard surface beneath. I also use the boards when drilling through things (hence the holes in the board and not my desk). 
The figures are now ready to be mounted on cardboard painting armatures and undercoated. I make my painting armatures - the thing you use to pick a figure up with when you are painting it - out of artist's mounting board. For 28mm infantry I use an armature measuring 20mm x 50mm, and for cavalry I use one measuring 25mm x 60mm. To glue the figure onto the armature I use a glue gun. A glue gun will secure a figure to the card quite firmly but it will be easy to remove cleanly when it is finished. I like my figures glued to one end. I tend to reuse armatures several times; the one above has only been used once before - you can see where the figure has been removed; when reusing an armature, always glue the figure to the opposite end to which it was glued the last time because the figure will come away easier when finished. I spray undercoat my figures black. Mostly I use Liquitex spray paint. This stuff is excellent. It dries matt and doesn't smell, so you can spray things outside in the winter (I always spray outdoors in a big lean to, open sided, shed) and bring them straight in doors when they are done to dry in the warm. Spray paint does not like cold weather. As it happens, I didn't use Liquitex for the figures below as I found half a rattle-can of old stock in a drawer and decided to use it up. It's not as good as the Liquitex because although it says matt on the tin it dries to a satin finish, and it stinks. It was warm and sunny today, so I left the smelly buggers outside to dry. 
I spray undercoat my figures in bulk where possible because I use just about the same amount of spray paint to undercoat fifty figures as twenty. This is a two battalion batch of fifty six figures. Two battalions worth of British Guards. I spray my figures on a sturdy board - in this case a large ceramic wall tile. 
Once sprayed I sort the troops out into batches and put them into paining trays. These trays allow me to move multiple batches to and from my painting desk with ease, so I can paint several different batches of figures on the same day. The painting tray should, ideally, have at least twice the footprint of the figures to allow them to be moved from one side of the tray to the other as each stage of the painting process occurs. In this case the tray is large enough to paint files or rows as the mood takes me but, because the figures are painted in procession I can be sure that every figure will get done. This is painting organisation at its simplest: Not having to go back to do missed figures saves a lot of time in the long run. I also take the trouble to mark figures of note. In this case the light and grenadier companies use the same figures (British Guards have full shako plates for both), so I daubed a spot of green on the lights and a spot of red on the grenadiers. I like doing batches of between 20 and 48 figures, preferring something mid range. A batch should ideally be large enough to allow some drying time - so that the first figure has dried before the last figure is done, usually 20 - 40 minutes - but not so large that the process becomes mind numbingly boring. I've done batches of 72 figures in the past and, although it is the quickest way to get figures done, it can be spirit crushing, especially on the third or fourth batch. In the case of my British Guards, I'm going to paint two batches of 28, one batch at a time.
I use a heavy blade, like a Stanley knife, to remove flash from the underside of a figure, and any other heavy flash. I use small pincers to remove 'sprues' connecting one bit of a figure (such as one connecting the bottom of a sword scabbard to the figure's base) before finishing with a scalpel** and files. Scalpels and needle files are also useful for cleaning up other small imperfections. Very rarely, especially where a mould has become worn and damaged, you get a heavier piece, or lump of flash. Where it's possible to reach it, I find a modelling drill the easiest way to remove the bulk of it, drilling it out, sometimes filling in the hole with putty (Milliput or Green Stuff) and reshaping the casting afterwards. Most commonly, I find this happens between a horses legs. Sometimes hands also need drilling to take flag staffs, lances or spears, and although a pin vice will do the job, a modelling drill will do it quicker. When you have one you will use it - so I advise any serious painter / modeller to buy one. I always clean figures with the figure pressed down on a small kitchen cutting board as it allows you to cut down and away from yourself to prevent injury without blunting the blade on any hard surface beneath. I also use the boards when drilling through things (hence the holes in the board and not my desk). 
The figures are now ready to be mounted on cardboard painting armatures and undercoated. I make my painting armatures - the thing you use to pick a figure up with when you are painting it - out of artist's mounting board. For 28mm infantry I use an armature measuring 20mm x 50mm, and for cavalry I use one measuring 25mm x 60mm. To glue the figure onto the armature I use a glue gun. A glue gun will secure a figure to the card quite firmly but it will be easy to remove cleanly when it is finished. I like my figures glued to one end. I tend to reuse armatures several times; the one above has only been used once before - you can see where the figure has been removed; when reusing an armature, always glue the figure to the opposite end to which it was glued the last time because the figure will come away easier when finished. I spray undercoat my figures black. Mostly I use Liquitex spray paint. This stuff is excellent. It dries matt and doesn't smell, so you can spray things outside in the winter (I always spray outdoors in a big lean to, open sided, shed) and bring them straight in doors when they are done to dry in the warm. Spray paint does not like cold weather. As it happens, I didn't use Liquitex for the figures below as I found half a rattle-can of old stock in a drawer and decided to use it up. It's not as good as the Liquitex because although it says matt on the tin it dries to a satin finish, and it stinks. It was warm and sunny today, so I left the smelly buggers outside to dry. 
I spray undercoat my figures in bulk where possible because I use just about the same amount of spray paint to undercoat fifty figures as twenty. This is a two battalion batch of fifty six figures. Two battalions worth of British Guards. I spray my figures on a sturdy board - in this case a large ceramic wall tile. 
Once sprayed I sort the troops out into batches and put them into paining trays. These trays allow me to move multiple batches to and from my painting desk with ease, so I can paint several different batches of figures on the same day. The painting tray should, ideally, have at least twice the footprint of the figures to allow them to be moved from one side of the tray to the other as each stage of the painting process occurs. In this case the tray is large enough to paint files or rows as the mood takes me but, because the figures are painted in procession I can be sure that every figure will get done. This is painting organisation at its simplest: Not having to go back to do missed figures saves a lot of time in the long run. I also take the trouble to mark figures of note. In this case the light and grenadier companies use the same figures (British Guards have full shako plates for both), so I daubed a spot of green on the lights and a spot of red on the grenadiers. I like doing batches of between 20 and 48 figures, preferring something mid range. A batch should ideally be large enough to allow some drying time - so that the first figure has dried before the last figure is done, usually 20 - 40 minutes - but not so large that the process becomes mind numbingly boring. I've done batches of 72 figures in the past and, although it is the quickest way to get figures done, it can be spirit crushing, especially on the third or fourth batch. In the case of my British Guards, I'm going to paint two batches of 28, one batch at a time.* For those who wondered how the strands of metal are formed in casting: When centrifugally casting, air pockets can sometimes prevent lead from reaching every nook and cranny. So, to let the air out, very fine holes are drilled through the mould (usually at right angles to the horizontal mould line) to let the air escape, and these holes fill with lead and thus the strands are cast onto the figure; the lead doesn't leak out of the mould because the holes are at 90 degrees to the metal flow, and so fine that the lead cools very quickly, self sealing, on entry into them. ** When it comes to 'modelling knives' I largely dispensed with them years ago. The blades are simply too expensive. I went over to surgical scalpels. I generally use a #3 handle and a #10A blade. Blades can be had for £9.99 for a box of 20 packets of 5. At 10p each you can throw blades away without thought when they get blunt - blunt blades cause more accidents than sharp ones - though I use blunt ones for scraping my painting palette clean (I have three handles).