How are SN1 and SN2 reactions carried out?

You could write whole chapters in an organic text to answer this question. And of course what follows is merely a guide to further reading. We know that #S_(N)1# reactions...#"substitution nucleophilic unimolecular"#, are in essence #"BOND-BREAKING"# reactions, that occur thru an intermediate, whereas #S_(N)2# reactions...#"substitution nucleophilic bimolecular"#, are in essence #"BOND-MAKING"# reactions.

And in the former, bond-breaking is the rate determining step...i.e. for an hydrocarbyl halide...

#R-Xstackrel("rate determining step")rarrunderbrace(R^(+) + Cl^(-))_"slow"#

The resultant carbocation...has a long-enuff lifetime to react with with an electron-rich nucleophile...say water or hydroxide ion...This mechanism is favoured by solvents that tend to solvate the individual ions, and thus solvents such as water, and alcohols tend to be employed in these reactions...

#R^(+) +HO^(-)stackrel("fast")rarrROH#

And, by contrast, in an #S_(N)2# reaction, BOND-MAKING between hydroxide, and the hydrocarbyl halide is the rate determining step...

#R-X +HO^(-)stackrel("slow")rarrROH+X^(-)#

The rate of reaction depends both on the concentration of hydroxide ion, and of the alkyl halide, (hence bimolecular kinetics), and importantly the reaction PROCEEDS with retention of any stereochemistry around the carbon centre...an optically active substrate will give rise to an optically active product given an #S_(N)2# mechanism... And in solvents such as ethers, the individual ions are not so strongly solvated, and as a consequence their nucleophilicity, and reactivity TEND to be enhanced.