Try to work quickly. This will come to you with practice. Cleanliness is important too. You can use clean water on cut surfaces.
Lop the top off the stock plant. Make sure that the cut is horizontal and that you can see the vascular bundle used for water and nutrient transport.
Save the head of the stock plant that was cut off so that it can be planted and used in future grafts.
You may need to trim (bevel) the edges of the stock so that the edges do not interfere with the portion to be joined when exposed surface dries and shrinks.
Slice a flat section of the graft plant, and trim the edges if you want to. Remember to make clean cuts. Don't crush the plant tissue when doing the cut.
Join the two plants, making sure the vascular tissues are as close to each other as possible.
Press down firmly to make sure there are no air pockets.
Secure the graft using rubber bands. Try not to apply too much pressure to avoid crushing plant tissue.
The rubber bands may be removed in a week's time. By that time too, the rubber bands will probably break, anyway.
The best description (and pictures) of grafting belongs to Miles Anderson's book. If you are serious about grafting, get his book -- it's a great book and he has great technique (of course, he's the guru of grafting).
Successes and Failures
My success rate for compatible stocks and grafts is about 70-80%. The grafting technique I use is available in the next section below. My list of successful grafts is as follows (I've lost count of the number of grafts, though):
Gymnocalycium mihanovichii cv. 'Hibotan' on Myrtillocactus geometrizans (about 8 specimens: good compatibility)
Gymnocalycium mihanovichii cv. 'Hibotan' on Hylocereus sp. (1 specimen)
Rebutia muscula on Myrtillocactus geometrizans (2 specimens: grafts offsetting well)
Melocactus sp. on Myrtillocactus geometrizans (1 specimen: growing well in hydroculture)
My list of unsuccessful grafts is as follows:
Gymnocalycium mihanovichii cv. 'Hibotan' on Echinopsis sp. (1 attempt: seems to have held but the Gymno discoloured slowly and later failed altogether)
Rebutia muscula on Echinopsis sp. (1 attempt: did not hold)
Myrtillocactus geometrizans on Opuntia sp. (1 attempt: graft turned purple and came off)
This list has not been updated for some time, so it wouldn't have included grafts done in 2000 and 2001.
Of course, there's probably not enough data to which is good and which is bad for certain, but it seems that Myrtillocactus geometrizans is an excellent stock. It's very easy to grow, and you can get it from a cheap packet of Horti mixed cactus seeds. But then I am using the Russian Space Agency standard -- one data point suffices .