Cloning DNA fragment essentially involves this steps:
isolation of the DNA of interest (or target DNA),
ligation,
transfection (or transformation),
screening/selection procedure.
DNA fragment to be cloned must be isolated, you do with PCR or restriction enzymes isolated from bacteria, which are able to recognize and cut specific sequences creating "Sticky" or "Blunt" DNA ends.
The plasmid is digested with restriction enzymes, opening up the vector to allow insertion of the target DNA. If the isolated DNA of interest and the plasmid or vector are digested with the same restriction enzyme, their sticky ends will be complementary. The two DNAs are then incubated with DNA ligase, an enzyme that can attach together strands of DNA with double strand breaks.
Following ligation, the recombinant DNA is placed into a host cell like E. Coli, in a process called transfection or transformation with high concentration of calcium or heat shocks.
Finally, the transfected cells are then cultured but some may not contain a plasmid with the target DNA because the transfection process is not usually 100% successful and the appropriate cultures must be selected with markers usually for antibiotic resistance. When the treated cells are plated on a petri dish of nutrient agar containing the antibiotic, only the rare transformed cells containing the antibiotic-resistance gene on the plasmid vector will survive.
Further analysis of the resulting colonies is required to confirm that cloning was successful. This may be accomplished by means of a process PCR or restriction fragment analysis, both of which need to be followed by gel electrophoresis and/or DNA sequencing. DNA sequence analysis, PCR, or restriction fragment analysis will all determine if the plasmid/vector contains the insert. Restriction fragment analysis is digestion of isolated plasmid/vector DNA with restriction enzymes. If the isolated DNA contains the target DNA, that fragment will be excised by the restriction enzyme digestion. Gel electrophoresis will separate DNA molecules based on size and charge.
Some suggestions:
Get a protocol from another investigator in the lab so you have skills and resources at your fingertips.
Many laboratory manuals are commercially available with simple and clear protocols provided for many fields, but you will need to refine the protocol yourself.
Methods section in published articles, last reliable place to find the protocol.