It's amazing how much more complex high school biology has become just since I graduated in 1990, we didn't even touch on these topics back in the 80s! I always find that when a textbook is vague about something, it's usually because the mechanism is poorly understood (or at least was when the text was written). This is always a difficult concept in and of itself to teach students--that in a biological process, steps 1, 2, 3, 6, 7, and 8 are fully understood, but steps 4 and 5 nobody has a clue about. Anyway, here's what I know: to make bacteria permeable to DNA you can use "electroporation", which involves exposing the cells to brief electric shocks of several thousand volts. These shocks do not kill the cells but make them temporarily permeable to DNA, it makes little holes in their membrane that the DNA plasmids can pass through. After the shock is stopped, the holes seal up and the membrane is unharmed.

There have been many different techniques developed to transfer DNA into different cell types too. Plant cells, yeast cells, even mammalian cells can be made to take up free DNA in a medium. The uptake mechanisms vary for each cell type. For example, mammalian cells will readily take up DNA when the DNA is converted to a fine precipitate by pre-treatment with Calcium phosphate.

Some strains of bacteria have a natural ability to take up naked DNA from surrounding media; however, the common lab strains (mostly E. coli) used in recombinant DNA technology methods do not have this property. Therefore, they must be treated a certain way to make them "competent" to be transformed with the recombinant DNA plasmid. (Note-it's the bacterial cell that is "transformed" by virtue of taking in the recombinant DNA plasmid). Now, the big question is: You have the DNA in a test tube along with some bacterial cells, what is the trick for getting the DNA in? The most common method for making the cells "competent" to take up DNA is to treat them with cold CaCl2 (calcium chloride). Another, more recently developed method is electroporation (you literally "zap" the cells with an electrical shock) while keeping the cells cold. Both methods tend to make the surface of the bacterial cell somewhat leaky and less "picky" about what can cross into the cell, but it is transient. As you might imagine, the properties of the buffer in which the cells are bathed during this time is crucial. The cold temperature probably does at least two things: (1) It prevents damage to the cell by protease and nucleases that might get activated by the various treatments (the enzymes don't work in the cold); and (2) it prevents the rapid "repair" of the cell surface, allowing enough time for the naked DNA (which you add to the tube of competent cells) to diffuse into the cells.

I think that you have several things kind of conglomerated. First off, you have to "get" a gene or a piece of DNA that you want to amplify. That piece of DNA is then put in a "vector", a small, circular piece of DNA that has a high copy number of replication in bacteria and the ability to be resistant to some type of antibiotic (typically ampicillin). How does your piece of DNA that you want get into the vector? Restriction enzymes. Your piece of DNA (linear) and the vector DNA (circular) are both cut with the SAME restriction enzymes and then put together in a reaction in the presence of ligase. The ligase will "ligate" [stick together] the pieces of DNA so that your DNA is now inserted in the vector to make a new circular DNA that has the vector and your DNA insert. Once you "make" that vector/your DNA hybrid (the recombinant plasmid), it needs to be put in bacterial cells that will overexpress it. You do that by using special E. coli cells that have been treated. Because of the treatment they go through, they are called "competent". You mix the DNA and the competent cells together and then heat-shock the cells. This will cause little holes in the bacteria, and the bacteria can then take up the DNA. Then you let the bacteria "recover" for about an hour, and then SELECT the bacteria with YOUR plasmid by using ampicillin in the growth media (that the cells grow in). The circular piece of DNA will have the genetic information to make the bacteria resistant to ampicllin (that would otherwise be killed by the
ampicillin).